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From: Jennifer P. Ogilvie and Kevin J. Kubarych, Multidimensional Electronic and Vibrational Spectroscopy: An Ultrafast Probe of Molecular Relaxation and Reaction Dynamics. In E. Arimondo, P. R. Berman, & C. C. Lin, editors: Advances in Atomic,

Molecular, and Optical Physics, Vol. 57, Burlington: Academic Press, 2009, pp. 249-321. ISBN: 978-0-12-374799-0

© Copyright 2009 Elsevier Inc. Academic Press.

Author’s personal copy

CHAPTER 5

Advances in Atomic, MoleISSN 1049-250X, DOI: 10.1

Multidimensional

Electronic and Vibrational

Spectroscopy: An Ultrafast

Probe of Molecular

Relaxation and Reaction

Dynamics

Jennifer P. Ogilviea and Kevin J. Kubarychb

aDepartment of Physics and Biophysics, University of Michigan,Ann Arbor, MI 48109, USAbDepartment of Chemistry, University of Michigan, Ann Arbor,MI 48109, USA

Contents 1. Introduction, Background, and Analogies 250

cular,016/S

and Optical Physics, Volume 57 # 2009 Elsevier Inc.1049-250X(09)57005-X All rights reserved.

1
.1 T imescales and Orders of Magnitude 251
1
.2 T he AMO Perspective: Photon Echoes,Ramsey Fringes, and NMR 252
1
.3 D iagrammatic Representation ofDynamical Evolution 258
1
.4 M olecular Perspective 265
2. T
wo-dimensional Electronic Spectroscopy 272
2
.1 Id iosyncrasies and Technical Challengesof Multidimensional ElectronicSpectroscopy 273
2
.2 E xperimental Implementations 273
2
.3 E xamples of 2D Electronic SpectroscopyExperiments 280
3. T
wo-dimensional Vibrational Spectroscopy 287
3
.1 Id iosyncrasies of MultidimensionalIR Spectroscopy 288
3
.2 E xperimental Implementation 289
249

250 Jennifer P. Ogilvie and Kevin J. Kubarych


3
.3 E xamples of Equilibrium 2DIRSpectroscopy 292
4. F
uture Directions 305
Ackno
wledgments 306 Refer ences 310
Abstract Multidimensional optical spectroscopy in the visible and infra-
red isa rapidlydeveloping techniqueenablingdirectobservationof complex dynamics of molecules in complex environmentssuch as liquids and proteins. Measuring the correlation betw-een excited and detected frequencies with sub-picosecondresolutionhasenabled the resolutionof long-standingproblemssuch as energy transfer in photosynthesis and the life-sustaining structural rearrangements of liquidwater. This chap-ter aims to provide a bridge between the concepts familiar inthe AMO physics community and how those ideas and experi-mental methods are applied to condensed phase molecularspectroscopy. We outline the technical challenges of thesepowerfulmethodswhile considering a few examples of experi-ments that showcase the unique perspective offered by 2Delectronic and vibrational spectroscopy.
1. INTRODUCTION, BACKGROUND, AND ANALOGIES

Detailed understanding of molecular processes in condensed phasesrequires spectroscopic probes of short length scales, providing ultrafasttime resolution with wavelengths ranging from the far infrared to theultraviolet. Following extensive progress in pushing temporal resolutionto the femtosecond level and below, the current state of the art of opticalspectroscopy of molecular dynamics in liquids, solutions, at surfaces, andin complex biological environments, is to extract the greatest possiblespectral information while maintaining the traditional time resolution ofultrafast spectroscopy. The approach discussed in this chapter uses meth-ods of photon echoes and hole burning to spread frequency informationon to two or more axes, reducing the spectral congestion that results fromstrong system–bath interactions. Since the first experimental demonstra-tions a decade ago (Hamm et al., 1998; Hybl et al., 1998; Lepetit & Joffre,1996; Likforman et al., 1997), multidimensional optical (visible andinfrared) spectroscopy has been applied to address many of the mostfundamental questions in condensed phase dynamics, and the majorityof that progress has been reviewed recently (Cho, 2008; Ganim et al., 2008;Hamm et al., 2008; Jonas, 2003; Wright, 2002; Zheng et al., 2007;

Multidimensional Electronic and Vibrational Spectroscopy 251


Khalil et al., 2003a). Viewed operationally, multidimensional spectros-copy appears to be nothingmore than an adaptation of established atomicand molecular optics techniques to the field of ‘‘chemistry.’’ The aim ofthis chapter, however, is to outline the points of contact and departurefrom the experimental methods and conceptual frameworks that under-pin the application of multipulse (or multifield) spectroscopy in differentregimes.

1.1 Timescales and Orders of Magnitude

In the condensed phase, electronic and vibrational spectral features oftenare not determined by the lifetime of the excited state. For example, thespectral width of the absorption of a common dye molecule, rhodamine6G, in ethanol solution is roughly 1400 cm�1 (41 THz, 170 meV), and itsbroad emission spectrum is �6000 cm�1 (170 THz, 745 meV). Despitethese very broad line widths, the excited state lifetime is 4 ns, fromwhich one would predict a wholly unphysical Lorentzian line width of0.0017 cm�1 (50 MHz, 207 neV). Clearly, the relevant system–bath inter-actions responsible for the spectral line widths are qualitatively distinctfrom those that determine the population relaxation.

Using traditional language, it is common to separate a spectral widthinto homogeneous and inhomogeneous contributions, leading naturallyto the pulsed photon echo technique or the frequency-domain hole-burning method to dissect the lineshapes experimentally. This approachhighlights a key feature of systems studied in the condensed phase:inhomogeneities are not static, and not all systems possess a single homo-geneous line width. The most widely used example of inhomogeneitywithin the context of atomic and molecular optics is Doppler broadening.In the absence of collisions during transit through the measurementapparatus, the distribution is static. In solution, however, a spectroscopi-cally observed molecule constantly interacts with its surroundings lead-ing to fluctuations of its eigenstate energies on a timescale that iscomparable to or longer than the ‘‘homogeneous’’ dephasing time. Theexperimental goal is therefore to determine the timescale for the stochasticsampling of the available microenvironments. Such measurements deter-mine the time taken by a spectroscopically ‘‘labeled’’ subpopulation torandomize, which often connects directly to the degree of system–bathcoupling, the magnitude of structural rearrangements and the extent ofexcitation delocalization. In this context, we refer to ‘‘labeling’’ in thesame way that a hole-burning experiment optically tags a subpopulationor transition. The timescale of frequency randomization varies, but fortypical condensed phase systems it is generally below a few picoseconds,and often it is subpicoseond. The experimental 2D data, unfortunately, donot uniquely determine an atomistic-level understanding, and it is widely



appreciated that theoretical and computational modeling plays a key rolein unpacking the spectral information content (Asbury et al., 2004c; Cho,2008; Cho et al., 2006; Cho & Fleming, 2005; Corcelli et al., 2004; Dreyer,2005a,b; Eaves et al., 2005a,b; Fecko et al., 2003; Fleming & Cho, 1996;Hanna & Geva, 2008a,c; Moran et al., 2003a,b; Paarmann et al., 2008;Schmidt et al., 2007; Skinner et al., 1981; Woutersen et al., 2002). Quantummechanical calculations are required to treat optical excitations and deter-mine transitions frequencies, but extended condensed phase systems areonly tractable using classical molecular dynamics simulations. Consider-able recent progress is ongoing to embed, self-consistently, a quantumsubsystem within a classical simulation, though this is an active area ofcurrent research (Hanna & Geva, 2008a,b,c).

1.2 The AMO Perspective: Photon Echoes, Ramsey Fringes,

and NMR

Borrowing some ideas from nuclear magnetic resonance, the problem ofinhomogeneous broadening has been addressed using nonlinear opticalspectroscopy. The Hahn echo, originally observed in nuclear spins (Hahn,1950), was adapted to atomic systems first in the case of ruby (Kurnitet al., 1964) and later was generalized to gases (Patel & Slusher, 1968).Figure 1 shows the stimulated photon echo pulse sequence along with the2 � 2 density matrix for a two-level system (2LS). The sequence has threetime intervals, t1, t2, and t3, where due to interactions with the three fields,

0

0

t1 t2 t3

Coherence Population Coherence

0

r00

0

00

r00

00

0 r11

00

0r10

00

0r10

0

0 0

r01

FIGURE 1 Pulse sequence and time labels for the three-pulse stimulated photon echo

used commonly in multidimensional spectroscopy. As an illustration, the time-

evolving density matrix is shown for a two-level system. During t1, the system evolves

in a coherence. The system is stored in a population on the ground or excited state

during t2, and the final field interaction induces a second coherence, which radiates a

signal that is detected in the phase-matched direction ks ¼ –k1 þ k2 þ k3



E1, E2, and E3, the system evolves alternately in coherences (rjk) andpopulations (rjj). What makes the sequence an ‘‘echo’’ is the phase conju-gation between the two coherence evolution periods. During t1 the den-sity is r01 and during t3 it is r10, thus effectively reversing the sign of timein the field-free Green function responsible for the system’s evolutionduring the second coherence interval. The lower set of density matricesshown during t2 and t3 indicate an alternative pathway where the systemreturns to the ground state following the second excitation pulse. Both theexcited state and ground state paths are required to evaluate the systemresponse to the applied fields. As is typically done in quantum optics, themacroscopic polarization is equal to the number density, N, of two-levelsystems multiplied by the expectation value of the dipole operator m, viz:

P t3ð Þ ¼ NTr r t3ð Þm½ �: (1)

Since the coupling between the external fields and the molecular system isgenerally weak, the density matrix is typically evaluated perturbatively tothird order, though for systemswhose low dimensionality permits it thereare nonperturbative schemes to evaluate the induced polarization andisolate the terms corresponding to the photon echo signal. It should benoted that in the perturbative regime, one typically considers the nthorder nonlinear susceptibility, w(n), and multidimensional spectroscopy isbased on the fact that the entire nth order susceptibility can be measured,including the phase. It is customary to define w(n)(o1, o2, . . ., on) in thefrequency domain, and the response function R(n)(t1, t2, . . ., tn) in the timedomain. Both quantities are tensors, and all aspects of the polarizations ofthe input and output fields are needed to compute the optical propertiesas well as to understand the underlying microscopic molecular processes.

The conventional interpretation of a photon echo experiment representsthe state of the systemusing apseudospin vector on theBloch sphere (Bloch,1946). By transforming to the rotating frame, each member of the inhomo-geneous ensemble precesses away from the origin at a rate proportional tothe difference between the specific frequency oi and the frame rotationfrequencyo0.When t2¼ 0, the second and third pulses flip the vectors lead-ing to the phase conjugation. The echo is emitted when the vectors precessfreely back to the origin, hence the gradual increase and decrease of theecho signal. Choosing t2 > 0 permits incoherent relaxation to occur beforethe refocusing pulse, providing access to lifetimemeasurements and obser-vation of spectral diffusion processes due to microscopic site randomiza-tion. Indeed, it is precisely the t2-dependence that provides much ofthe chemical information in condensed phase photon echo experiments.



The most commonly implemented strategy to obtain multidimensionaloptical spectra is based on the photon echo sequence. Since the goal isspectral resolution, rather than determining dephasing rates by changingthe time delays, the information one seeks is generally in the frequencydomain. Thus, the emission frequency information is obtained by measur-ing the emitted signal field in a spectrometer by interferometric combina-tion with a fully characterized reference local oscillator field (Dorrer et al.,2000; Gallagher et al., 1998; Lepetit et al., 1995; Lepetit & Joffre, 1996;Likforman et al., 1997), which is typically derived from E3 (Figure 1).A Fourier transform algorithm can be used to isolate the electric fieldsusing spectral interferometry (Lepetit et al., 1995). Given this ability tomeasure the signal field as a function of o3, the excitation frequency infor-mation is obtained by scanning the t1 delay and then Fourier transformingthe recorded spectra along this dimension, yielding o1. The result ofthe above procedure is a two-dimensional spectrum correlating the excitedfrequency o1 with the detected frequency o3 at a specific value of theso-called ‘‘waiting time’’ t2. Further discussion of the details of obtaining2D lineshapes that correspond to absorptive processes are described below.

As an illustration using a model of alternatively homogeneously andinhomogeneously broadened 2D spectra, Figure 2 shows simulations of1D and 2D spectra for a two-level system (Faeder & Jonas, 1999). The 2Dspectrum of a 2LS with exponential dephasing is a 2D Lorentzian, and thenet 2D spectrum of an ensemble of 2LSs is the sum of the individual 2Dspectra. The two models produce similar 1D absorption spectra using thesame size ensembles. The two 2D spectra, however, show a clear differ-ence in the shape of the spectrum. For the homogenous model there isessentially no specific dependence of the emission frequency on theexcitation frequency, whereas the inhomogeneous model exhibits a clearcorrelation between excitation and emission frequencies. The spectralcorrelation, whose signature is the pronounced elongation along thefrequency diagonal (o1 ¼ o3), directly reflects the underlying inhomoge-neous distribution. The asymmetry of the 2D lineshape as seen in theslices (Figure 3) provides a measure of the degree of inhomogeneousbroadening. In the discussion so far, dynamics in the system is onlycontained in the spectral lineshape. As will be discussed below, someof the most chemically relevant information is obtained by observinghow the 2D lineshape asymmetry relaxes as the waiting time is increased(Cho, 2008; Faeder & Jonas, 1999; Kwak et al., 2008b; Lazonder et al.,2006). This kind of information is one of the key distinctions betweenhow the photon echo pulse sequence is used in multidimensional spec-troscopy and how it is used in atomic physics.

Another experimental method commonly employed in traditionalAMOphysics is the production of Ramsey fringes with spatially separatedfields (Ramsey, 1950). First demonstrated in the microwave spectral

0.8

1

0.6

0.4

A (w

) (a

.u.)

0.2

0

0.2

0.1

−0.1

−0.2

−0.3

0.2

0.1

0

−0.1

−0.2

−0.3−0.3 −0.2 −0.1 0.1 0.2

0

0

1

0.8

0.6

0.4

A (w

) (a

.u.)

0.2

0−0.5 0

w−w0

w3−w

0

w3−w

0

w1−w0

−0.3 −0.2 −0.1 0.1 0.20w1−w0

0.5 −0.5 0w−w0

0.5

FIGURE 2 Model 2D spectra for an ensemble of two-level systems with (left)

primarily homogeneous broadening, and (right) inhomogeneous broadening. The

inhomogeneous broadening leads to a distinctive frequency correlation parallel to the

o1 ¼ o3 diagonal. For most systems of interest in condensed phase dynamics, this

inhomogeneity is transient and as the waiting time between excitation and detection is

increased, the 2D lineshape asymmetry relaxes



domain, Ramsey fringe spectroscopy (Figure 4) has been adapted to manysituations using both linear and Doppler-free, two-photon absorptionapproaches, temporal pulses, and additional fields (Ramsey, 1995).From the system’s perspective, the coherence induced through the firstinteraction is read out by a second field whose effective time delay isdetermined by the transit time between the fields. For an atomic beam atfinite temperature the Maxwell–Boltzmann distribution of molecularspeeds in the z-direction imparts a static inhomogeneity that is manifestedas a distribution of transit-time broadening. The phase acquired for anatom moving at a velocity vi, with center frequency oi in transiting thedistance Dz between the two fields is oiti, where ti ¼ Dz/vi. As a functionof the laser frequency o0, which is tuned through the resonance of theatoms, fringes develop in the resulting fluorescence emission measuredafter the last field. The spectral resolution achievable with the Ramseyfringe method is inversely proportional to the distance between the fields.Clearly the method can only function in systems where the dephasingtime—the time for the decay of the initially produced coherence—is much

1

0.8

0.6

0.4

0.2

0−0.5 0

A(w

) (a

.u.)

0.5w1–w0

1

0.8

0.6

0.4

0.2

0−0.5 0

A(w

) (a

.u.)

0.5w1–w0

FIGURE 3 Slices through the simulated 2D spectra shown in Figure 2. The blue curves

show diagonal slices, and the green curves show the antidiagonal slices passing

through the peak maximum. The difference in line width is a measure of the

inhomogeneity



longer than the transit between the two fields. In solution or in liquids,there is no analogous experiment, since diffusion is the limiting transportprocess. One might imagine, however, that mass-selected clusters may beable to be studied using a Ramsey fringe method in the gas phase. Despitethe clear analogy between Ramsey fringes and photon echoes, again theinhomogeneity is static, and the vast majority of the formal theoreticaldevelopments of both Ramsey fringes and photon echoes are for staticsystems of two- or three-levels with very weak coupling to the environ-ment (i.e., via spontaneous emission). Thus the similarities aid in makingtranslations between the molecular questions of condensed phase dynam-ics and isolated atoms, but the differences highlight the motivation to usemultidimensional spectroscopy as a probe of dynamics in complexenvironments.

w

t1 t2

FIGURE 4 Schematic diagram of a Ramsey fringe experiment where spatially

separated fields excite a beam of atoms traveling from left to right. By tuning the laser

frequency o, the inhomogeneous transit-time broadening of the thermal atomic beam

can be removed



It has been customary to draw analogies between multidimensionalspectroscopy and multidimensional NMR spectroscopy (Scheurer &Mukamel, 2002). Despite many parallels in experimental implementationand data processing, however, the key strengths of multidimensionalspectroscopy are manifest most clearly when the analogies fail. Morespecifically, whereas a set of coupled spin-1/2 particles gives rise to a2N-dimensional space of system eigenstates, a simple pair of coupledoscillators gives rise to an infinite dimensional state space. Furthermore,important information regarding the anharmonic nature of the underly-ing potential surfaces is contained in transitions involving states otherthan those in the ground and first excited vibrational exciton bands.Calculating the transition frequencies and transition dipoles in the 2Dspectra from first principles is far more challenging than in the case ofNMR. Spins also interact only very weakly with their environments, sothat although the emitted signals are vanishingly small, there is virtuallyno penalty for increasing the number of pulses in the sequence, enablingthe observation of correlations between many different kinds of nuclei.Finally, the existence of a nonlinear optical signal is conditioned by theanharmonic nature of the Hamiltonian. In the case of vibrations, anhar-monicities often constitute a relatively small fraction of the overall transi-tion frequencies; they may overlap spectrally with the fundamentaltransitions thereby diminishing the signal. In electronic systems, thereare often overlapping and unintuitive excited states. Such complicatedenergy level structure does not exist in the case of NMR. The intrinsic time



resolution of NMR is orders of magnitude lower than optical 2D spectros-copy, which is necessarily performed with ultrafast (<100 fs) pulses.Thus, from a practical viewpoint multidimensional spectroscopy can beused as an ultrafast probe of triggered, nonequilibrium chemical eventson the femtosecond to picosecond timescale (Baiz et al., 2008; Bredenbecket al., 2004, 2005, 2007; Cervetto et al., 2008; Chung et al., 2007; Kolanoet al., 2006; Smith & Tokmakoff, 2007b).

1.3 Diagrammatic Representation of Dynamical Evolution

Most textbooks on quantum and nonlinear optics describe the systemresponse to a series of pulses using a two-level system and encode thefour density matrix elements in three coordinates (Allen & Eberly, 1987).An applied field may be viewed as imparting a torque, O, on the three-dimensional vector, s, as:

ds tð Þdt

¼ O tð Þ � s tð Þ: (2)

By inserting exponential relaxation of the inversion r11–r00 with a decaytime given by T1 and coherences, r12 and r21, given by T2, realistic modelsof atomic dynamics can be used to predict and interpret multipulseexperiments. It is common to obtain results for the generated signal bydirectly integrating the equation of motion for the Bloch vector, leading tofamiliar effects of free-induction decay, Rabi oscillations, optical nutation,electromagnetic-induced transparency, and photon echoes (Allen &Eberly, 1987). For situations where the inversion is small, due to largedetuning from atomic resonance or because the fields are weak, the time-dependence may be evaluated perturbatively by expanding the systemresponse in powers of the applied field (Borde, 1983). The perturbativetreatment naturally leads to a finite-order signal that appears at a well-defined spatial location (ks¼�k1� k2� k3� � � �) and emission frequency(os ¼� o1 � o2 � o3 � � � �). For the purposes of enumerating the possiblefield-matter interaction events and field-free evolution periods, one oftengraphically represents the given term in the perturbation sum (Berman &Lamb, 1970). Shown in Figure 5, we adopt the convention commonly usedin the 2D community and show an energy level diagram where fieldinteractions are represented as vertical arrows, with solid arrows indicat-ing that the interaction raises or lowers a bra-side density matrix element,while the dashed arrows indicate transitions made to the ket side (Ulnesset al., 1998). Choosing a particular signal direction fixes the wave vectorsof the exciting fields. A diagrammatic method helps to list exhaustively allevolution pathways by representing the density matrix as two verticallines and the field interactions as arrows that point up and to the right for

|b⟩

|a⟩

|0⟩

|b⟩

DiagonalGSB

ks= - k1+ k2+ k3 ks= +k1- k2+ k3

DiagonalSE

Cross-peakESA

DiagonalGSB

DiagonalSE

Cross-peakESA

|a⟩

|0⟩

|b⟩

|a⟩

|0⟩

|b⟩

|a⟩

|0⟩

|b⟩

|a⟩

|0⟩

|b⟩

|a⟩

|0⟩

FIGURE 5 Double-sided Feynman diagrams (top) for rephasing (ks¼ –k1þ k2þ k3) andnonrephasing (ks ¼ þk1 – k2 þ k3), with the associated wave-mixing energy level

diagrams for amodel two-manifold, three-level system. For the first three fields, ket- and

bra-side interactions are denoted with dashed and solid arrows, respectively. The

emission of the signal field is denoted by a thick solid arrow. Abbreviations used are

ground state bleach (GSB), stimulatedemission (SE), andexcited state absorption (ESA)



positive wave vector, and up and to the left for negative wave vector(Borde, 1983). In analogy with other diagrams used to sum perturbationseries, these diagrams have come to be called ‘‘double-sided Feynman’’diagrams. To enumerate all paths, all permutations are required such thatthe arrow direction and time ordering is preserved. For example, a fieldwith wave vector of �k2 contributes by inducing a downward transitionon the left (ket side), and an upward transition on the right (bra side). Theonly constraint is that the system must terminate in a population after theemission of the signal to ensure a nonzero trace. With these rules, it isstraightforward to list all possible pathways to nth order provided theenergy levels and transition dipole moments connecting the levels areknown.

Figure 5 shows double-sided Feynman diagrams with the correspond-ing energy level diagrams for a three-manifold, three-level system, whichserves as a good model for both a single anharmonic oscillator or a typicalvisible dye molecule. These diagrams are valid when the applied electricfield can only couple states that are in adjacent manifolds. The horizontaldotted line shows the expected location of the second excited state if thesystem were assumed to be harmonic. Since the field amplitude duringemission from jbi to jai has an opposite sign to that emitted from jai to j0i,in a harmonic system the two contributions destructively interferecompletely. Although there are two pathways emitting ato0a and only oneemitting atoba, the harmonic scaling of the transitionmoment mba ¼

ffiffiffi2

pma0



leads to a net weight for the induced absorption that is twice that of theground state bleach signal jma0j2jmbaj2 ¼ 2jma0j4. The nonlinear signal’sexistential dependence on anharmonicity in the molecular potentialmakes the spectroscopy particularly sensitive to precisely those aspectsof the Hamiltonian that linear spectroscopy cannot reveal.

To provide a general description of the types of 2D lineshapes that havebeen observed in several experimental contexts, Figure 6 shows a sche-matic spectrum corresponding to the three-level system depicted inFigure 5. The slanted peak with solid contours illustrates the spectrumdue to the ground state pathways such as ground state bleach (GSB) andstimulated emission (SE), whereas the dashed contours, which haveopposite sign, indicate the excited state absorption (ESA). In most of thevibrational systems that have been studied, the ESA slant appearsroughly parallel to the ground state contribution, indicating that theinhomogeneity does not substantially affect the anharmonicity. Theremay be systems, however, where the correlation between the 0–1 andthe 1–2 transitions is more complicated, leading to qualitatively differentground state and excited state lineshapes. In electronic spectroscopy,there are not as simple universal trends in the relationship between thefirst and second excited manifolds, and the inhomogeneity of ESA has notyet been explored experimentally.

wexcite

wde

tect

FIGURE 6 Cartoon 2D spectrum for a three-manifold, three-level system showing the

ground state (solid) and excited state (dashed) contributions to the signal. The diagonal

elongation indicates inhomogeneous broadening. In a harmonic system, the positive

ground state and negative excited state signals overlap spectrally, thus interfering

destructively



Systems with multiple states in the first excited manifold exhibit acharacteristic pattern of cross-peaks in a 2D spectrum in the same waythat such peaks are observed in 2D NMR spectra. The simplest modelwhere cross-peaks arise is shown in Figure 7 for a two-manifold,three-level system. There are two related phase-matching conditionstermed rephasing (kR ¼ –k1 þ k2 þ k3) and nonrephasing (kNR ¼ þk1 –k2 þ k3). The significance of these two directions is discussed belowregarding the 2D spectral lineshape. Here, we briefly mention that inthe ‘‘rephasing’’ geometry, the system evolves at conjugate frequenciesduring the two coherence periods and can therefore produce an ‘‘echo.’’In the ‘‘nonrephasing’’ case, system evolution occurs at the same fre-quency during both coherence periods and, therefore, cannot produce amacroscopic rephasing. We point out the two sequences to highlight thedifferences in the peak locations in a system with strong coupling leadingto cross-peaks. For the rephasing sequence there are two paths that lead toa given cross-peak, and two that lead to a given diagonal peak. In thenonrephasing sequence, however, three paths lead to diagonal peakswhile only one leads to the cross-peak. It is noteworthy that the pathsevolving as coherences contribute differently in their locations in therephasing and nonrephasing sequences (Khalil et al., 2004; Nee et al.,2008). The rephasing coherence paths involve excitation at one frequencyand emission at the other frequency, whereas the nonrephasing coherencepaths are to due excitation and detection at the same frequency. Thus, themodulation that is due to these coherences appear on the cross-peaks forthe rephasing sequence and on the diagonal for the nonrephasingsequence (Engel et al., 2007; Khalil et al., 2004; Nee et al., 2008; Nemethet al., 2008, 2009; Tekavec et al., 2009).

A schematic 2D spectrum for themultilevel system depicted in Figure 7is shown in Figure 8. Although the level diagrams of Figure 7 suppress thesecond excited state manifold for simplicity, this cartoon spectrumincludes the cross-peaks due to ESA (Golonzka et al., 2001). The cartoonshows two diagonal peaks with their corresponding red-shifted ESApeaks of opposite sign, as well as two cross-peaks (solid contours) andthe associated cross-peaks (dotted contours) that are due to ESA. Bymeasuring the relativemagnitudes of these peaks it is possible to constraina coupled Hamiltonianmodel of the underlying eigenstates, including therelative orientations of the transition dipoles (Golonzka et al., 2001).

1.3.1 Causality and the Absorptive Lineshape

Due to the obvious causality of the linear optical response that followsoptical excitation, there is a simple relationship between absorption anddispersion via the Kramers–Kronig relations. In higher order response

ks= -k1+ k2+ k3

ks= +k1- k2+ k3

|b⟩|a⟩

|b⟩|a⟩

|0⟩

|0⟩

|b⟩|a⟩

|b⟩|a⟩

|0⟩

|0⟩

Diagonal Cross-peak Diagonal Cross-peak

|b⟩|a⟩

|b⟩|a⟩

|0⟩

|0⟩

|b⟩|a⟩

|b⟩|a⟩

|0⟩

|0⟩

Diagonal Cross-peak Diagonal

Coherence

Coherence

Coherence

Coherence

FIGURE 7 As in Figure 5, double-sided Feynman diagrams and wave mixing energy

level diagrams for a two-manifold, three-level system. The paths leading to coherences

during t2 are labeled



wexcite

wde

tect

FIGURE 8 Cartoon 2D spectrum for a three-manifold, three-level system. The solid

diagonal and cross-peaks arise from the pathways shown in Figure 10. The additional

dashed and dotted peaks are due to excited state absorption. 2DIR spectra typically

show both diagonal (dashed) and off-diagonal (dotted) anharmonicity



functions the relationship is less transparent (Faeder & Jonas, 1999). Obtain-ing absorptive lineshapes is of particular interest since the often uninterest-ing dispersive effects of density and thermal gratings can complicate theextraction of the response due solely to resonant absorption and emission.Separating absorptive spectra fromdispersive contributions alsomaximizesthe information content of a 2D spectrum, allowing discrimination of sig-nals such as ESA and GSB that have different relative signs. It has beenrecognized that the real part of the photon echo signal does not contain thepurely absorptive 2D spectrum unless two different experiments arerecorded where the roles of the first and second pulses are interchanged(Hybl et al., 2001b; Khalil et al., 2003b). In the conventional echo sequence,the signal is detected in the rephasing (kR ¼ –k1 þ k2 þ k3) direction. Toobtain an absorptive 2D spectrum, it is necessary to also measure the signalthat emerges in the nonrephasing (kNR ¼ þk1 – k2 þ k3) direction and addthis to the rephasing signal. Drawing analogies toNMR spectroscopy, Jonasand coworkers have explained the reasoning behind the requirement ofmeasuring both rephasing and nonrephasing pathways (Faeder & Jonas,1999). Consider the sequence of pulses in the 2D measurement as depicted

t1

Scan

Scan

−t1

k1 k1k2

k1 k3k2

t1

“−t1”

k2 k3k1

k3

FIGURE 9 Illustration of causality in multidimensional spectroscopy. If the system

were to exhibit no dynamics following the second excitation, for an arbitrary waiting

time it would be possible to obtain an absorptive spectrum by scanning the pulse

with wave vector k1 from positive to negative time delays, followed by Fourier

transformation of the signal with respect to t1. Instead,most systems do have dynamics

and the waiting time is not arbitrary. To obtain an absorptive spectrum the pulse

ordering must be reversed. In the pump-probe geometry the pulse sequence is

inherently symmetric, obviating the need to scan two delays



in Figure 9. Prior to excitation by the third pulse, the system is in a popula-tion; causality dictates that the 2D signalwe seek should be zero, implying aKramers–Kronig relationship with respect to the time variable t3 (assumingdelta function pulses). However, a similar causal relationship does not holdwith respect to the time variable t1, during which the system is in a coher-ence. Here, a negative t1 delay corresponds to an inversion of the timeordering of the first two pulses, with signal being present at positive andnegative t1 values, regardless of the pulse ordering. Thus, collecting thesignal in a single phase-matching direction for a fixed value of the waiting



time t2 amounts to truncating the signal at t1¼ 0, resulting in amixing of realand imaginary components in the Fourier transform with respect to t1. Toavoid this problem, symmetric recording of the signal with respect to t1 isneeded, permitting a complex Fourier transformover the full range�1< t1< 1 (Faeder & Jonas, 1999; Hybl et al., 1998). Obtaining this full range tosuccessfully separate absorptive and dispersive spectra can be achievedexperimentally using a number of different approaches that will be dis-cussed in further detail in Section 2. We note that for large values of t2 if thesystem evolution is slowly varying, a scan over positive and negative t1delays in a single phase-matched direction should suffice for collecting bothrephasing and nonrephasing pathways, as illustrated in Figure 9.

Depending on the experimental approach used for obtaining rephasingand nonrephasing signals, an additional step may be required to unam-biguously assign the absorptive and dispersive components of the 2Dspectrum. This ambiguity arises unless the absolute phase of the radiatedsignal is known. Since many experimental implementations measure thecomplex rephasing and nonrephasing signals by spectral interferometrywith a reference field, only phase differences are directly measured,leaving an undetermined phase offset. The solution that is commonlyused is to rely on the projection slice theorem (Faeder & Jonas, 1999),which states that the real part of the projection of the 2D spectrum alongthe o1 axis is equal to the spectrally resolved pump-probe signal (Hyblet al., 2001b). This method requires a separate acquisition of the spectrallyresolved pump-probe signal for the appropriate t2 delay. Several alterna-tive approaches have recently been developed to measure the absolutephase directly (Backus et al., 2008a; Bristow et al., 2008). Alternately, thepump-probe implementation of 2D spectroscopy avoids this problemaltogether and will be discussed in Section 2.2.

1.4 Molecular Perspective

Since nth-order multidimensional spectroscopy essentially involves themeasurement of the nth-order optical response function, there are clearlymany parallels with the experiments and formalism developed withinthe context of AMO physics. Here we outline the types of molecularphenomena that are particularly well suited to study with multidimen-sional spectroscopy.

1.4.1 Coupling

Any spectroscopic technique measures transitions between energy eigen-states, and these eigenstates do not always correspond to spatially localizedexcitations. The inherent freedomgranted by quantummechanics to choose



any convenient basis often allows one to use a basis that is structurallyintuitive. Upon choosing a N-dimensional basis, it is possible to expressthe potential energy in the Hamiltonian in terms of the N coordinatesexpanded about a reference configuration q0 ¼ q01; q

02; . . . ; q

0N

� �:

V q1; . . . ; qNð Þ ¼XNj;k¼1

@2V

@qj@qkjq0ðqj � q0j Þðqk � q0kÞ

þXNj;k;l¼1

@3V

@qj@qk@qljq0ðqj � q0j Þðqk � q0kÞðql � q0l Þ þ . . . ;

(3)

where the first sum of terms includes the harmonic ( j ¼ k) contributionas well as the bilinear ( j 6¼ k) coupling. A nonzero nonlinear opticalresponse requires terms in the potential beyond second order, and thelowest of these appear in the second sum. The diagonal anharmonicity( j ¼ k ¼ l) accounts for the anharmonicity along each of the basiscoordinates, whereas the off-diagonal anharmonicity accounts for cou-pling through the anharmonicity. In principle, an nth-order spectros-copy can only measure terms to nth-order in the potential, but theeigenenergies will generally be sensitive to higher order terms as wellas the finite basis size in any practical calculation. Provided the systempermits sufficient spectral isolation for a reasonable number of spectro-scopic modes, it is possible to determine the parameters of the potential.In general, coupling between transitions appears as cross-peaks in the2D spectrum, and one may extract the parameters of the potential usingthe energies and relative amplitudes of the peaks (Bredenbeck & Hamm,2003; Ding et al., 2005; Golonzka et al., 2001; Woutersen & Hamm, 2001).

The model described above for a nominally vibrational problem can beeasily adapted to treat the multiple chromophores in strongly coupledelectronic systems. In molecular aggregates, branched polymers, quan-tum wells and natural photosynthetic protein assemblies, the electronicdegrees of freedom are often coupled by the strong interactions betweenneighbors (Axt & Mukamel, 1998). The strong interactions lead to split-tings in the electronic eigenstates. In close analogy to the strong couplingin vibrational states in molecules, the specific eigenenergies are especiallysensitive to distance and orientations of the constituent units. Thusspectroscopic information is also structurally sensitive, though the linkbetween spectrum and structure is often difficult to establish quantita-tively. Both electronic and vibrational multidimensional spectroscopycan elucidate the underlying degree and nature of coupling, as well asthe sensitivity to the environment and its fluctuations (Cho, 2008).



1.4.2 Line Broadening

Line broadening has long been understood as resulting from the effects oflifetime and frequency fluctuations. In contrast to the case of isolated gasatoms, in the condensed phase line widths are rarely determined byexcited state life time due to the overwhelming significance of environ-mental fluctuations. Considering the system–bath interactions to producea time-dependent instantaneous energy gap expressed as a frequencyo ¼ DE=�h:

o tð Þ ¼ hoi þ do tð Þ; (4)

the absorption lineshape can be viewed as a consequence of the time-correlation function of the frequency fluctuations (Kubo, 1969):

C tð Þ ¼ hdo tð Þdo 0ð Þi: (5)

Within the Gaussian–Markov model for fluctuations, fast fluctuationsdecay exponentially, leading to Lorenztian lineshape functions, whereasslow fluctuations decay as a Gaussian, corresponding to Gaussianspectral lines. Such a model permits smooth transit from homogeneous(Lorentzian) to inhomogeneous (Gaussian) limits of spectral features.As described above, a linear spectrum in the condensed phase lackingperfectly isolated spectral features cannot distinguish unambiguouslybetween these two microscopically distinct underlying dynamics (Loring& Mukamel, 1985; Tanimura & Mukamel, 1993). Using the 2D lineshape,however, it is possible to determine more rigorously the dynamical originof a broadened line.

1.4.3 Orientation

Condensed phasemolecules seldom rotate freely; instead the orientationalevolution is diffusive. Since the nonlinear response function is fundamen-tally a correlation function of the dipole operator, using fields with differ-ent polarization combinations it is possible to isolate both the orientationbetween coupled transitions as well as the orientational relaxation of thewhole molecule (Tokmakoff, 1996). Neglecting for the moment dynamicalreorientation, the nonlinear optical signal results from four light-matterinteractions mediated by the dipole operator. A peak in a 2D spectrum atthe location (oa, ob), where both states are in the first excitation manifoldfor simplicity, is weighted by the following product of transition dipolemoments: m!0am

!a0m

!0bm

!b0. Assuming an isotropic distribution of molecular

orientations, the ratio of this cross-peak to the (oa, oa) and (ob, ob)diagonal peaks may be used to deduce the orientation of the 0!a and



0!b transitions. The orientations of the transitions may be associatedwith structural constraints provided one has a structural correspondencewith the spectroscopically bright eigenstates (Hochstrasser et al., 2002;Wang et al., 2006).

Since there are four total field polarizations in a third-order experi-ment, it is also possible to use different combinations of laser polariza-tions and selection of signal polarization by an analyzer to enhance theorientational information in a 2D spectrum. Although there is initialpreferential excitation of molecules whose transition moments are paral-lel to the laser polarization direction, as these molecules diffusively reori-ent, the generated signal will decrease. Thus, by increasing the waitingtime (t2) delay, orientational dephasing will lead ultimately to a rerando-mized sample. The timescale for orientational diffusion is often measuredusing transient absorption anisotropy. Measuring the pump-probesignal with parallel (DAk) and perpendicular (DA?) polarizations, theratio (DAk � DA?)/(DAk þ 2DA?) characterizes the anisotropy of thetransient absorption signal. Since transient absorption spectroscopy canbe derived from a full 2D spectrum, this information is naturallycontained within 2D spectra (Tokmakoff, 1996).

1.4.4 Coherence

The first and third time periods (t1 and t3) are often referred to as coher-ence periods because in all cases the system must evolve as a coherence(off-diagonal density element) to produce a third-order signal. There is norequirement, however, that the system evolve in a population (diagonaldensity element) during the so-called ‘‘population’’ time t2. Indeed, thevery short pulses used to perform impulsive, Fourier transform 2D spec-troscopy with the photon echo sequence are typically broad in frequencyand able to excite many different transitions. When two or more transi-tions are accessible within the laser bandwidth, the second interactionmay connect the ground state with a different excited state. The sequencecan be represented with the density matrix for a three-level system as:

r00 0 00 0 00 0 0

0@

1A!

m0a

0 r0a 00 0 00 0 0

0@

1A!

mb0

0 0 00 0 00 rba 0

0@

1A!

ma0

0 0 00 0 0rb0 0 0

0@

1A!

mb0;

(6)

where the system is excited at the frequency o0a and emits at o0b. Duringthe population time, the system is in the coherence rba, thus the a!0transition probability will oscillate at the frequency Oba ¼ o0b – o0a. Thesecoherences have in fact been observed since the development of femtosec-ond pump-probe spectroscopy, but their appearance at specific locations in



the 2D spectrum enables more direct assignments of the constituent eigen-states (Khalil et al., 2004; Nee et al., 2008). The time dependence of coher-ences can provide additional information about the frequency–frequencycross-correlation involving pairs of states. Coherence is a generic propertyof nonlinear spectroscopy and will be discussed below in several contextsof purely electronic, vibronic and purely vibrational transitions.

1.4.5 Spectral Diffusion

Hole-burning double resonance spectroscopy has enabled isolation ofthe homogeneous line width hidden within an inhomogeneous band byburning a hole and probing the resulting loss of absorption. In a transienthole-burning experiment, a moderately narrow band laser excites a smallsubpopulation and a time-delayed broadband probe interrogates the sam-ple; the absorption difference in the presence and absence of the pump ismonitored. In a statically inhomogeneous system, the transient hole doesnot change in time. If the system is dynamically inhomogeneous, however,the hole will fill in as the initially excited molecules adopt different con-formations or experience different local solvent environments. Since thefilling in of the hole is analogous to how a concentration hole fills in bydiffusion, the stochastic sampling of all energetically accessible substates isreferred to as spectral diffusion (Klauder & Anderson, 1962). Multidimen-sional spectroscopy provides a direct probe of spectral diffusion since adynamically inhomogeneous systemwill display adiagonally elongated 2Dlineshape at short waiting times, followed by a loss of this initial correlation(Figure 10). As the correlation is lost, the 2D lineshape will become lessasymmetric, and the decay of the asymmetry provides a direct measureof spectral diffusion (Hybl et al., 2001a; Lazonder et al., 2006).

Increasewaitingtime

wexcite wexcite

wde

tect

wde

tect

FIGURE 10 Cartoon representation of spectral diffusion. At short waiting time (left) the

signal displays inhomogeneity as a diagonal elongation. With increased waiting time

the spectrum relaxes to a symmetrical 2D shape as each initially excited subpopulation

stochastically samples all the available local environments or conformations



Spectral diffusion is a sensitive measure of solvation dynamics and slowstructural rearrangements, allowing a spectral window into the dynamicalrigidness or floppiness of hydrogen bonded complexes (Elsaesser et al.,2007), liquids (Cowan et al., 2005; Eaves et al., 2005a; Zheng et al., 2007;Kraemer et al., 2008) and key locationswithin proteins (Bandaria et al., 2008;Fang et al., 2006, 2008; Lim et al., 1998).

1.4.6 Chemical Exchange

It is a cornerstone of chemical dynamics that at equilibrium the rate offormation of the product is equal to the rate of formation of the reactantsuch that the overall ratio of both species’ concentrations maintains aconstant value: the equilibrium constant. Using transition state theory, it ispossible to derive an expression for the forward and backward rate con-stants, whose ratio yields the equilibrium constant. The values of the rateconstants are determined by the relative energy differences of the reactant,product, and the transition state. For transition state barriers that aresufficiently low, the transit of a molecule from reactant to product, andvice versa, may bemonitored using ultrafast spectroscopywhile the systemremains essentially at equilibrium (Cahoon et al., 2008; Fang et al., 2006;Finkelstein et al., 2007; Ishikawa et al., 2008; Kim & Hochstrasser, 2005a,2006; Kim et al., 2008; Kwak et al., 2008a; Woutersen et al., 2001; Zheng &Fayer, 2007; Zheng et al., 2005, 2007). An extension of the ideas discussedin the context of spectral diffusion—where there is not assumed to be anenergetic barrier between the substates—is that of ultrafast chemicalexchange. Figure 11 shows a schematic chemical exchange measurement,indicating the loss of inhomogeneity as well as the growth of exchangecross-peaks. By exciting a sample of species A and B that are in equilib-rium, at early waiting time before appreciable numbers of molecules havereacted, the 2D spectrum appears as a simple mixture of two componentswhose proportions determine the equilibrium constant. As the waitingtime is increased, more reactants become products and more productsbecome reactants. Initial excitation at oA of the reactant leads to emissionatoB of the product. At the same time, initial excitation atoB of the productleads to emission at oA of the reactant. By monitoring the growth of theexchange cross-peak, it is possible to determine the rate of interconversion.More specifically, the t2-dependent growth of the cross-peak yields thesum of the forward and reverse rate constants (Perrin &Dwyer, 1990). Theindividual rate constants can be determined provided one knows theequilibrium constant, which can be obtained from steady-state measure-ments. By repeating the experiment at different temperatures, the Arrhe-nius law provides an estimate of the energy gap between the transitionstate and the reactant and product. The key experimental requirement isthat the signal persists long enough to observe the interconversion,

wexcite wexcite

wde

tect

wde

tect

Increasewaiting

time

FIGURE 11 Cartoon representation of chemical exchange measured with 2D

spectroscopy. Two species are in dynamic equilibrium, and at early waiting time show

two uncoupled, isolated spectral features. With increased waiting time the lineshape

asymmetry relaxes and the populations initially labeled become scrambled due to the

equilibrium fluctuations between the two species. Growth of the cross-peaks indicates

chemical exchange, with a rate equal to the sum of the forward and reverse rates.

The timescales accessible using 2D optical and vibrational spectroscopy are naturally in

the ultrafast regime, enabling the direct measure of activated processes with very low

energy barriers



limiting the method to particularly fast reactions or to those which havevery long lived vibrational markers. Additionally, it remains a question towhat extent the vibrational excitation itself modifies the equilibrium or thekinetics of exchange. Chemical exchange is an established technique in thetoolbox of NMR spectroscopy, and its recent extension to the IR illustratesthe generality of the experimental approach. Although there is no a prioriprohibition against electronic chemical exchange spectroscopy, electronictransitions typically are even more poorly characterized as near-equilibrium than are their IR counterparts. Additionally, structuralchanges such as complexation to form a dimer often lead to quenching ofthe electronic excited state. Nevertheless, small electronic chromophoreprobes attached to a larger macromolecular complex may prove practicalin observing large-scale equilibrium fluctuations of the underlyingsupport.

1.4.7 Energy Transfer

There are few useful generalities about the microscopic details of energytransfer in condensed phase systems. Both vibrational and electronicenergy can transfer according to an array of mechanisms, not all ofwhich can be understood simply using Fermi’s Golden Rule. Clearly thesystem studied must possess a spectral marker for energy transfer, andprovided one exists, the waiting time dependence of a 2D spectrumenables one to map the disappearance of excitation energy in one spectral



region to the appearance at another. In vibrational systems spatial corre-lations can be extracted by exploiting the identification of spectral signa-tures with specific structural components. Electronic energy transferwithin or between chromophores must be analyzed using quantumchemical calculations since the spectrum-structure link is often indirect(Cho et al., 2005), though the rich history of ultrafast transient absorptionoffers many examples of electronic energy relaxation.

The following sections will focus separately on the specific implemen-tations of multidimensional spectroscopy in the visible spectrum to studyelectronic transitions and in the infrared to probe molecular vibrations.Though there are certainly many conceptual and practical differences, theunderlying theoretical and conceptual framework is common to both, andadvances in the field have seen immediate application in both spectralregimes wherever possible.

2. TWO-DIMENSIONAL ELECTRONIC SPECTROSCOPY

The physical nature of the condensed phase often leads to highly inho-mogeneously broadened electronic spectra. Examples of this are many:a dye molecule (chromophore) in solution experiences a unique microen-vironment that reflects the local configuration of solvent molecules. Alight-harvesting complex, consisting of an arrangement of chromophoresin different protein environments with inter-chromophore distances andorientations fixed by the protein architecture, has broadband absorptionthat suits its function. In its Fourier transform implementation, the com-bination of high time and frequency resolution makes 2D electronicspectroscopy (2DES) a valuable tool for studying energy and chargetransfer and chemical dynamics in the condensed phase. This combina-tion is particularly powerful for studying inhomogeneously broadenedsystems where spectral changes reflecting the dynamics of the chromo-phore and its environment occur on rapid timescales. In this respect 2DESoffers distinct advantages over single molecule spectroscopies thatcurrently suffer from limited time resolution.

In contrast to the numerous theoretical and experimental 2DIR studies,work at visible frequencies has been lacking. This can in part be attributedto the relatively more difficult experimental implementation of 2D spec-troscopy at higher frequencies. Since the early work of the Jonas group(Faeder & Jonas, 1999; Hybl et al., 1998, 2001a,b, 2002; Jonas, 2003), 2DEShas being applied to a growing number of problems ranging from thestudy of solvation dynamics (Hybl et al., 2002; Jonas, 2003), to energytransfer in photosynthesis (Brixner et al., 2005; Engel et al., 2007; Readet al., 2008; Zigmantas et al., 2006) and in semiconductor systems (Borcaet al., 2005; Li et al., 2006; Yang et al., 2007; Zhang et al., 2007). Parallel



efforts to simplify experimental implementations of 2DES are ongoing andaim to extend 2DES to new frequency regimes such as the ultraviolet. Herewe discuss some of the experimental challenges of 2DES and outline twomethods that have been used to meet these challenges. We concludewith several examples from the recent 2DES literature that highlight thecapabilities of 2DES.

2.1 Idiosyncrasies and Technical Challenges of

Multidimensional Electronic Spectroscopy

The pulse sequence for 2DES, shown in Figure 1 reveals some of thechallenges of its implementation at optical frequencies. Being a Fourier-transform-based spectroscopy, interferometric precision of �l/100,corresponding to timing errors of 0.017 fs at 500 nm is needed to produceaccurate Fourier transform frequencies (Jonas, 2003). In the infrared, therequirement is considerably easier to meet: at 5 mm the same degree ofprecision requires timing errors of less than �0.17 fs. Even at infraredfrequencies, scanning artifacts from imperfect translation stages candegrade the quality of the spectra (Ding et al., 2006; Volkov et al., 2005).In a well-constructed interferometer, optical pathlength fluctuations andmechanical instabilities introduce timing errors of �0.1 fs over a 20-minperiod ( Jonas, 2003), limiting the duration of possible experiments. Theseproblems suggest that mechanisms for both stabilizing the phase andprecisely measuring time delays are needed for successful 2DESexperiments.

The broad features of electronic spectra can be somewhat sharpened bythe separation of 2D spectra into absorptive and dispersive components.This separation requires knowledge of the complex signal field, andtherefore contains phase information that can distinguish between signalcontributions of opposite sign, such as ESA and stimulated emissionsignals. As discussed in Section 1.4.1, obtaining the absorptive spectrumrequires detection of the signal field during the second coherence time t3and collection of two different phase-matched signals: the rephasing andthe nonrephasing contributions. In general, the detection of the signalfield can be readily achieved by spectral interferometry with a referenceelectric field (Lepetit & Joffre, 1996). The proper separation then requiresinterferometric stability between signal and reference fields, andknowledgeof the absolute signal phase.

2.2 Experimental Implementations

2D spectroscopy has been implemented using a number of different meth-ods, and each of them solves the problems of timing precision, phasestability and lack of knowledge of the absolute phase in different ways.



Fully noncollinear measurements are desirable for two reasons: to permitisolation of weak signals from strong excitation pulses, and to facilitateselection of a subset of density matrix elements ( Jonas, 2003; Mukamel,1995). Fully collinear measurements, closer to their NMR cousins, havebeen developed by the Warren group using pulse-shaping methods (Liet al., 2007; Tian et al., 2003; Wagner et al., 2005). Here phase-cyclingrather than phase-matching is used to separate the desired signal compo-nents. Fully collinear methods have the advantage of operating in therotating frame, reducing the required sampling rate. They are also lessrestricted by the sample size than the fully noncollinear implementationsthat require coherent buildup of signal. However, the collinear geometryis not background free, requiring 16 different phase settings to isolate theabsorptive 2D spectrum. The Marcus group has developed a collinearsetup for fluorescence-based 2DES (Tekavec et al., 2007). While theyemploy standard interferometers, they avoid problems associated withinterferometer stability by using a phase modulation method that decou-ples pulse-pair timing errors from their relative phase (Tekavec et al.,2006). More recently a hybrid, partially noncollinear approach has beenadopted by a number of groups (DeFlores et al., 2007; Grumstrup et al.,2007; Myers et al., 2008; Shim et al., 2007; Tekavec et al., 2009). In thefollowing sections we focus on two different experimental approaches to2DES, providing details of both methods and highlighting the strengthsand weaknesses of each implementation.

2.2.1 Diffractive Optics

A simplified solution to implementing 2D Fourier transform spectroscopyis to use diffractive optics (DO) to generate the required pulse sequenceand phase-matched geometry. DO have been used previously for imple-menting fifth-order Raman spectroscopy (Astinov et al., 2000; Kubarychet al., 2003) and transient-grating studies (Goodno et al., 1998; Daduscet al., 2001; Maznev et al., 1998) where they have permitted passivelyphase stable heterodyne detection. To extend their use to 2D spectros-copy, the ability to implement a time delay between pulse pairs wasneeded. This was done by refractive delays that vary the optical path-length by rotating an arrangement of coverslides (Cowan et al., 2004;Ogilvie et al., 2002a) or translating a wedge pair (Brixner et al., 2004).In condensed phase systems at room temperature, the first coherenceperiod t1 is very short-lived, with typical dephasing rates that are oftencommensurate with the pulse duration (�20–50 fs), meaning that refrac-tive delays are practical for scanning such short durations. Cowan et al.(2004) also showed that reflective optics could be used to exploit correla-tions in phase errors between pulse pairs to achieve overall phase errorcancellation (Cowan et al., 2004; Ogilvie et al., 2002a).



2DES experiments are frequently implemented using amplified lasersystems, which offer stable and reliable output with appropriate pulseenergy and repetition rate for studies of dynamical processes over a broadrange of timescales. Our diffractive-optics-based experimental apparatusfor 2DES is shown in Figure 12(a). A Titanium sapphire oscillator seeds a1 kHz regenerative amplifier to produce 40-fs pulses with an energy of1 mJ at 800 nm. To permit access a broader range of wavelengths, from thevisible to the near IR, we divide the output from the regenerative ampli-fier and pump two noncollinear optical parametric amplifiers (NOPA).These NOPAs produce �10 mJ, 20 fs pulses over the broad range from�475–1000 nm (Wilhelm et al., 1997). Having independently tunablepump and probe pulses allows two-color 2DES (2C2DES), which addsflexibility to previous implementations of 2DES by providing access to abroader range of electronic transitions, spanning the visible and near-infrared. We generate the excitation pulse sequence for the experiment bysending the pump and probe beams into a passively phase-stabilizeddiffractive optics arrangement (Cowan et al., 2004; Ogilvie et al., 2002a).The standard BOXCARS geometry is used, as shown in Figure 12(b).Three of the four beams constitute the excitation pulse sequence, whilethe fourth beam, which conveniently propagates in the signal direction, isreduced in intensity and sent through the sample prior to the other pulsesto act as a reference field for heterodyne detection of the signal. Thepassive phase-stability of the DO approach is derived from the fact thatthe first-order diffracted beams from the DO provide beams 1 and 2 fromthe first NOPA and 3 and 4 from the second NOPA. By passing throughalmost identical optical pathways between the DO and the sample, timingerrors caused by mechanical instabilities are common to the beams hitting

Oscillator Time delay

Wedgepair

Referenceand signal

Sample

At sample

13

PM

24

t2

t1

k2

k1

k3

PC

DO

NOPA2

NOPA1

Regenerativeamplifier

Spectrometer

(a) (b)

FIGURE 12 (a) Two color 2DES based on the diffractive optics approach. NOPA:

noncollinear optical parametric amplifier, PC: prism compressor, PM: parabolic mirror.

(b) Phase-matching geometry



the same optical elements. Using refractive delays we have demonstrateda high degree of phase stability (l/90) between the relevant pulse pairs,leading to a high degree of phase stability in the signal (Cowan et al.,2004). To scan the t1 delay while maintaining this passive phase stabilitywe use a refractive delay line as previously demonstrated, employingpairs of wedges (Brixner et al., 2004) rather than a tilted coverslip config-uration (Cowan et al., 2004). We calibrate this delay by scattering pulses 1and 2 from a pinhole and performing spectral interferometry, achievingtiming precision of several attoseconds. Two pairs of wedges allow us toscan positive and negative values of t1, collecting the rephasing andnonrephasing signals to separate absorptive and dispersive componentsof the 2D spectra (Jonas, 2003). We use a standard delay line for the t2delay where interferometric precision is not needed. We spectrally dis-perse our heterodyned signal in a spectrometer acquiring the signalelectric field through spectral interferometry (Lepetit & Joffre, 1996).

Other implementations of fully noncollinear 2DES have solved thephase stability problem by using active phase stabilization (Zhang et al.,2005). The Nelson group has pioneered the use of pulse shapers for fullynoncollinear 2DES (Gundogdu et al., 2007; Vaughan et al., 2007), employ-ing a 2D pulse shaper to provide both the phase-matched beam geometryand excellent control of the timing, phase and spectral content of the pulses.Similar to Cowan et al. (2004), a recent implementation by Brixner et al.(2004) (Selig et al., 2008) employs the idea of using pulse pairs arranged toexploit phase error correlations to maintain overall phase stability. Unlikethe Cowan method, diffractive optics are not employed in this method,permitting easier extension to broadband applications.

While the DO approach to 2D spectroscopy solves the phase stabilityproblem, it does not provide absolute phase information since it employsspectral interferometry, which only measures the relative phase betweensignal and reference fields. By acquiring the spectrally resolved pump-probe signal for each t2 delay, the projection slice theorem can be used toextract the absorptive 2D spectrum (Faeder & Jonas, 1999). Alternately,two new approaches have recently been developed that solve the absolutephase problem optically without the need for a supplemental pump-probe measurement (Backus et al., 2008a; Bristow et al., 2008).

2.2.2 The Pump-Probe Geometry

It was pointed out by Jonas and coworkers that a pump-probe geometrywould remove some of the technical challenges of 2D spectroscopy(Faeder & Jonas, 1999). With collinear pump beams, rephasing and non-rephasing signals are emitted in the same direction, allowing their simul-taneous collection and eliminating the need for separate scans that canintroduce phase errors. This approach was identified several years before



the introduction of 2D optical spectroscopy (Cho et al., 1992), but the fullpower of the method was only recently recognized and properly imple-mented first using a germanium acousto-optic pulse shaper (Shim et al.,2007), and later using a Mach–Zehnder interferometer (DeFlores et al.,2007). In the case of Zanni’s implementation, a pulse shaper was used toproduce the collinear pump pair for excitation. Later Zanni extended thework to the near-IR (Grumstrup et al., 2007), demonstrating the method inrubidium. Pulse-shaping has the advantage of providing excellent controlof the timing of the pump pulse pair and permits phase-cycling opportu-nities that are unavailable with interferometer-generated delays. Being apump-probe measurement, the transmitted probe acts as a heterodyningfield with well-defined timing with respect to the signal. In scatteringsamples, phase-cycling can be used to discriminate against scattered lightand improve signal quality (Shim et al., 2007).

Ogilvie et al. (2002a) have implemented the pulse-shaping method atvisible frequencies, with a configuration shown in Figure 13(a) (Myerset al., 2008). The pump beam was sent into an acousto-optic pulse shaper(Dazzler, Fastlite) to create the first two excitation pulses with a variabledelay. Unlike a delay produced by an interferometer, the pulse shaperallows the introduction of an arbitrary carrier wave phase shift thatpermits phase-cycling schemes analogous to those used in NMR(Li et al., 2007). The modulation applied to produce the pulse pair wasof the form |E(o)|(1 þ exp[i(ot1 þ ’12)]) where E(o) is the spectralamplitude of the pulse and ’12 is the relative carrier wave phase shift.Pump and probe pulses are crossed at the sample cell at a small angle(�2�). The heterodyne-detected 2D signal was spectrally resolved at1 kHz, providing the n3 axis of the 2D spectrum. The t1 delay was scannedusing the Dazzler, as further detailed in reference Myers et al. (2008)

Spectrometer

Probe andsignal

Sample

t1 t2

t2

t3

Sapphire

(a) (b)

k3

k2 k1

Oscillator

Regenerativeamplifier

Dazzler™NOPA

FIGURE 13 (a) 2DES in the pump-probe geometry, using a pulse shaper to generate

the pump pulse sequence. NOPA: noncollinear optical parametric amplifier, PC: prism

compressor, PM: parabolic mirror. Here a continuum probe pulse, generated by

focusing into a sapphire plate is used. Alternatively a second NOPA can provide the

probe pulse. (b) Phase-matching geometry



In the pump-probe geometry, the total signal intensity contains thesignal of interest as well as a host of other signals, including the transmit-ted probe light, the free-induction decay from pulse 3, and the pump-probe signals resulting from the pair-wise interactions of the probe withpump pulse 1 and pump pulse 2 (Faeder & Jonas, 1999). The desired third-order signal P

3ð ÞS2D can be partially isolated from the other contributions by

chopping every other pump pulse with a mechanical chopper or with thepulse shaper and then computing the difference between consecutive spec-tra. Alternatively, a phase-cycling scheme, where consecutive pump pulsesare given a’12¼ 180� relative phase shift, produces signals that are oppositein sign at every second laser shot. Compared to chopping, this schemeeffectively doubles the duty-cycle of the experiment. The signal of interestis then:

S o3; t2; t1ð Þ / � Im E3 o3ð ÞP 3ð Þ

S2D o3; t2; t1ð Þh i

/ Re E3 o3ð Þ R Rð Þ o3; t2; t1ð Þ e�if12 þ R NRð Þ o3; t2; t1ð Þ eif12

n oh i:

(7)

Because pulses 1 and 2 are essentially interchangeable, for f12 ¼ f1 �f2 ¼ 0� the time domain signal must be symmetric with respect to t1 ¼0 andmust therefore be purely real (Faeder & Jonas, 1999). If f12¼ 90�, thesignal becomes antisymmetric with respect to t1 ¼ 0 and all of the signalamplitude will be present in the imaginary part. In practice, since we onlyscan positive t1 values, care must be taken when computing and inter-preting the final 2D spectrum. We can either symmetrize the data prior toFT along t1, or take the cosine transform (when f12 ¼ 0�) or the sinetransform (when f12 ¼ 90�) with respect to t1. Along the t3 dimension,causality requires that there be no 2D signal for t3 < 0. Upon applying thesymmetry and causality conditions, Fourier transform with respect to t1and t3 yields the complex 2D spectrum, the real part of which is purelyabsorptive.

As recent work has shown, separating rephasing and nonrephasingspectra can be useful for observing vibronic modulation of 2DES line-shapes (Nemeth et al., 2008), as well as for analyzing the joint frequencyfluctuations or dephasing time of coupled chromophores (Cheng &Fleming, 2008; Ge et al., 2002). Although these signals are collected simul-taneously in the pump-probe geometry, phase-cycling can be used toseparate these contributions by collecting data with f12 ¼ 0� and f12 ¼90� (Myers et al., 2008). A demonstration of the separation is shown inFigure 14, where the rephasing and nonrephasing contributions areshown for a simple laser dye, LDS750 in acetonitrile. At room temperatureLDS750 exhibits a large Stokes shift due to a combination of solvation and

v 3 (

TH

z)

v1 (THz)

v1 (THz)

465

450

435Re

Im

420

405520 535 550 565 580

v 3 (

TH

z)

465

450

435

420

405520 535 550 565 580

v1 (THz)

v1 (THz)

465

450

435

420

405520 535 550 565 580

465

450

435

420

405520 535 550 565 580

R(R) R(NR) Sum

v 3 (

TH

z)

v1 (THz)

v1 (THz)

465

450

435

420

405520 535 550 565 580

v 3 (

TH

z)

465

450

435

420

405520 535 550 565 580

FIGURE 14 Separation of the rephasing and nonrephasing contributions for LDS750 in

acetonitrile at t2 ¼ 500 fs. The rows contain real and imaginary spectra as indicated. Left

column: components of the rephasing signal. Middle column: components of the

nonrephasing signal. Right column: addition of the rephasing and nonrephasing

signals to give the absorptive spectrum (top right)



intramolecular relaxation processes (Kovalenko et al., 1997). This Stokesshift is evident in the two-color 2D spectrum. We note that the imaginarypart of the data shown in the bottom right resembles a dispersive spec-trum. However, it contains no new information, being equivalent to aKramers–Kronig inversion of the absorptive data over a finite frequencyrange and is therefore not a complete measure of the dispersive suscepti-bility (Albrecht et al., 1999).

Two-dimensional spectroscopy in the pump-probe geometry is consid-erably easier to implement than noncollinear approaches. The greatbenefit of using a pulse shaper to generate the t1 delay is that it removesany uncertainty in the location of t1 ¼ 0, allowing easy measurement ofthe absorptive 2D spectrum (Grumstrup et al., 2007; Shim et al., 2007). Theinformation content of the absorptive component has been shown tobe equivalent to that obtained in the noncollinear geometry (Faeder &Jonas, 1999). Use of a pulse shaper provides access to phase-cyclingprocedures that can improve the signal-to-noise ratio (SNR) (Shim &Zanni, 2009; Shim et al., 2007) and isolate signals of interest. Anothersignificant benefit of the pump-probe geometry is the straightforwardextension to 2C2DES, and the use of a continuum probe, allowing the



exploration of coupling between electronic transitions over a broad fre-quency range. Ultrabroadband continuum-probed 2DES has recentlybeen demonstrated by the Ogilvie group (Tekavec et al., 2009). Whilesimpler to implement than noncollinear approaches to 2D spectroscopy,the pump-probe geometry is not a background free measurement and assuch will have a poorer SNR. As in any pump-probe experiment, the SNRcannot be optimized because the relative intensity of the signal andtransmitted probe (which acts as the heterodyning reference field) cannotbe independently controlled. For some of the tensor components, it hasbeen shown that polarization schemes can remove this difficulty, allow-ing signal to noise that should approach that of background-free measure-ments (Myers et al., 2008; Xiong & Zanni, 2008).

2.3 Examples of 2D Electronic Spectroscopy Experiments

2.3.1 Energy Transfer in Light-Harvesting Systems

The light-harvesting systems responsible for photosynthesis in plants andbacteria exploit a common architecture that resembles an energy ‘‘fun-nel.’’ Photons are absorbed by light-harvesting complexes consisting ofantennae arrays that surround a reaction center, to which the solar energyis transferred and stored as stable charge separation (Blankenship, 2002;van Amerongen et al., 2000). The high efficiency with which this energytransfer occurs (>95%) has generated intense interest in these systemsand a desire to understand their design principles for use in artificialdevices. In light-harvesting complexes, the geometric arrangement oflight absorbing pigments and the local protein environment tune theabsorption properties and direct the flow of energy. What makes themparticularly challenging spectroscopic subjects is their mixture of disor-der induced by different conformations and microenvironments and theelectronic coupling of the closely spaced pigments (van Amerongen et al.,2000). The effects of coupling and disorder produce broad electronicspectra that are difficult to interpret. Mapping the energy and chargetransfer pathways in photosynthetic systems and understanding theunderlying design principles are part of an ongoing effort for which2DES is uniquely suited to provide (1) the high time resolution necessaryto follow the early events of energy transfer; (2) a more direct view ofelectronic coupling; and (3) the ability to dissect the inhomogeneouslybroadened lineshapes to separate disorder and electronic coupling effects.

Figure 15 depicts a simple case of two coupled chromophores exchang-ing energy and describes how the coupling and energy transfer process isobserved with 2DES. Two extremes of the strength of dipole–dipolecoupling are considered. When the coupling is weak, excitations arelocalized on the individual chromophores and energy transfer can be

(a)

(b)

t2= 0

J12

2

1

0

Exciton

Levels

Strong coupling

Weak coupling

w1w2

wdetect

wexc

w2

w1

w1 w2

t2> 0

wdetect

wexc

w2

w1

w1 w2

t2= 0

wdetect

wexc

w2�

w1

w1 w2

t2> 0

wdetect

wexc

w2

w1

w1 w2

FIGURE 15 (a) Energy level structure and cartoon 2D spectra for a two chromophore

system in the case of weak coupling. At t2 ¼ 0 no cross-peaks are present in the 2D

spectrum,while at later waiting times the gradual appearance of a cross-peak below the

diagonal indicates incoherent energy transfer (incoherent ‘‘hopping’’). (b) Energy level

structure and cartoon 2D spectra for a two chromophore system in the case of strong

coupling, where an excitonic picture is appropriate. At t2 ¼ 0 cross-peaks indicate

excitonic coupling via a common ground state. As a function of the waiting time t2 thecross-peaks carry the information about energy transfer. By recording 2D spectra at

different t2 values, the dynamics and pathway of energy flow can be mapped out. A 1D

spectroscopymeasurement would integrate over theoexc axis of the 2D spectra leading

to overlapping contributions that complicate interpretation of the data. In particular, the

cross-peaks that provide information about coupling and energy transfer may be

buried beneath other contributions



described by Forster theory as an incoherent hopping between chromo-phores. In the 2D spectrum, as the waiting time t2 is increased, thisappears as an increase in the amplitude of both the lower energy diagonalpeak and the cross-peak below the diagonal, indicating net populationflow into the lower energy transition, often referred to as ‘‘downhill’’energy transfer. In the strong coupling limit an excitonic picture is used,where excitations are delocalized. In this case, as a function of waitingtime t2 the cross-peaks in the 2D spectrum oscillate in amplitude at thedifference frequency of the two exciton states.

The Fleming group has used 2DES to study a number of light-harvesting systems, some of which has been recently reviewed by Cho(2008). They have focused much of their work on the Fenna–Matthews–Olson complex (FMO) (Brixner et al., 2005; Cho et al., 2005; Engel et al.,2007; Read et al., 2008): a pigment/protein structure in green sulfurbacteria that serves as bridging structure to transfer energy betweenlarger peripheral antennae and the reaction center. The structure of a



single FMO subunit is shown in Figure 16, which illustrates its arrange-ment of seven bacteriochlorophyll molecules. Its relatively small numberof chromophores and well-characterized structure has made FMO a goodmodel system for studying energy transfer in photosynthesis (Camara-Artigas et al., 2003). Previous work had proposed an excitonic model ofenergy transfer (Cho, 2008; Vulto et al., 1999). This model derived theexciton states from linear spectroscopic measurements and the knowncrystal structure to estimate site energies for the different pigments, andelectronic coupling between them. An excitonic model combined withtransient absorption measurements permits kinetic modeling of the exci-ton state populations to derive a picture of energy flow. However, the t2dependence of the cross-peaks in a 2D spectrum allows a more directvisualization of excitonic coupling and energy transfer. The intuitivepicture of energy flow through FMO that was previously derived fromnonlinear spectroscopy involved a stepwise downhill flow of energy intothe lowest exciton state 1 (Cho, 2008; Vulto et al., 1999). While consistentwith an overall downhill flow, the 2D spectroscopy of the Fleming grouprefined the model, revealing two dominant energy transfer pathways inFMO. These pathways often skip between states that are adjacent inenergy, instead choosing pathways with stronger spatial overlap of theexcitonic wavefunctions, as depicted in Figure 16.

In their work on the FMO complex, the Fleming group also observedelectronic coherences. These coherences are generated by using excitationpulses with large enough bandwidth to excite multiple electronic transi-tions and are manifested as modulations in the amplitude of diagonal and

EnergyExciton

7

6

54

3

2

1

12, 4

5, 6

3, 7

FIGURE 16 The seven bacteriochlorophyll pigments of FMO (left), and the excitonic

level structure (right). Gray shading indicates the delocalized nature of some of the

excitons. The two dominant energy transport pathways are depicted in red (dashed)

and green (solid). Structure taken from the Protein Data Bank, 1M50 (Camara-Artigas

et al., 2003). Figure adapted from Brixner et al. (2005)



cross-peaks as a function of the waiting time, as depicted in the Feynmandiagrams of Figure 7 and discussed in Section 1.4.4. The frequency of themodulations observed in FMO was consistent with the expected excitondifference frequencies used to model the 2D spectra. While the work wasperformed at low temperature (77K), the coherences were remarkablylong-lived, raising intriguing questions about both the role of the proteinarchitecture in preserving the electronic coherence and the possible impor-tance of coherence in maximizing energy transfer efficiency (Engel et al.,2007). Studies on other natural light-harvesting systems at temperatures ashigh as 180K have shown similar coherences (Lee et al., 2007), as havepolymers at room temperature (Collini and Scholes, 2009).

In addition to natural photosynthetic systems, candidates for artificiallight-harvesting systems are also beginning to be addressed with 2DES.Kauffmann and coworkers have recently studied bi-tubular J-aggregates,examining energy transfer between the inner and outer walls (Nemethet al., 2009; Sperling et al., 2008), while the Fleming group has also studiedexciton dynamics in a different J-aggregate system (Stiopkin et al., 2006).

2.3.2 Vibrational Wavepacket Dynamics in 2DES

In pump-probe spectroscopies utilizing ultrashort pulses, the incidentpump pulse may excite a coherent superposition or ‘‘wavepacket’’ ofvibrational states, as in impulsive-stimulated Raman scattering (ISRS)(Merlin, 1997; Mukamel, 1995; Yan &Nelson, 1987a,b). Many experimentshave explored vibrational wavepacket dynamics in systems as diverse assemiconductors to proteins (Armstrong et al., 2003; Mukamel, 1995; Zhuet al., 1994). On the excited state, vibrational wavepackets are generatedbecause the excited state geometry is different from that of the groundstate; in acquiring the new excited state geometry, the molecule under-goes structural rearrangements that drive vibrational modes. The band-width of the excitation pulses determines the highest vibrationalfrequency that can be excited impulsively; a 10-fs pulse is capable ofexciting modes as high as 1500 cm�1. In a 2DES experiment, vibrationalwavepackets can also be generated, where after the interaction of thesecond pulse the system evolves in a coherence between vibrational stateson the ground or excited electronic state, as depicted in the Feynmandiagrams in Figure 7. The vibrational wavepacket motion will be mani-fested as modulations in the 2D spectra as a function of the waiting time.

Recently Nemeth et al. (2009) have explored the effect of vibrationalwavepacket motion on 2D spectra, studying the laser dye N, N0-bis(2,6-dimethylphenyl)perylene-3,4,9,10-tetracarboxylicdiimide (PERY, Figure 17(a)) dissolved in DMSO (Nemeth et al., 2008). This dye has been studiedextensively by other groups using a broad range of nonlinear optical tech-niques (Larsen et al., 2001; Ohta et al., 2001). A distinguishing feature of

(a)

(c) (d)

(b)

PeryCH3 H3C

H3CCH3

O

O

O

O

600

332 fs 460 fs 587 fs 707 fs

1

0.8

0.6

0.4

0.2

Inte

nsity

(a.

u.)

Dia

g./a

nti-d

iag.

wid

th

0

1.3

1.25

1.2

1.15

1.1

1.05

1

0.95200 300 400

t2 (fs)

500 600 700

500 520

332 fs 460 fs

587 fs707 fs

540

Frequency (THz)

560 580 600

580

560

v 3 (T

Hz)

v1 (THz)

540

520

500

550 565 580

v1 (THz)

550 565 580

v1 (THz)

550 565 580

v1 (THz)

550 565 580

N N

FIGURE 17 (a) Structure of PERY. (b) Absorption (solid) and emission (dotted) spectra of PERY showing the pump (dashed) and continuum

(dash-dotted) spectra. (c) Absorptive 2D spectra for PERY in DMSO at different waiting times t2. (d) Ratio of diagonal to antidiagonal widths

for the central peak (solid blue), lower peak (dashed red), and upper peak (dotted green), showing modulation with a �240-fs period. Also

indicated are the t2 values corresponding to the 2D spectra shown in (c)




PERY is the progression of a high frequency vibrational mode (1410 cm�1)that strongly modulates its absorption and emission properties, shown inFigure 17(b). While most spectroscopic investigations of PERY have usedpulse durations too long to impulsively excite the 1410 cm�1 mode, lowerfrequency intramolecular modes, particularly one at 139 cm�1, have beenseen in threepulse echopeakshift (3PEPS), transient-grating studies (Larsenet al., 2001; Ohta et al., 2001), and the 2Dwork of the Kauffmann group.

The Ogilvie group has recently studied PERY in DMSO using thepump-probe geometry with a continuum probe to span the broadbandabsorption and emission features (Tekavec et al., 2009). Figure 17(b)shows the pump and probe wavelengths used in their experiments,while Figure 17(c) shows 2D spectra of PERY in DMSO taken at differentwaiting times (Tekavec et al., 2009). Three distinct peaks are present in the2D spectra located at n3 ¼ 605 THz, n3 ¼ 560 THz, and n3 ¼ 520 THz. Thecentral peak arises from a combination of bleach of the ground state andStokes-shifted excited state emission, leading to a slight shift below thediagonal. The low frequency cross-peak arises from the double-peakedemission characteristic of PERY’s vibronic progression. The origin of thehigh frequency cross-peak is the common ground state between the 0! 0and 0 ! 1 transitions. Although subtle, a modulation in the ellipticity ofthe peaks is evident, as shown in Figure 17(d) where we plot the ratio ofdiagonal to antidiagonal full-width half-maximum (FWHM) of the threepeaks as a function of waiting time. All three peaks exhibit ellipticityoscillations with a 240-fs period, in accordance with 3PEPS and transient-grating studies (Larsen et al., 2001; Ohta et al., 2001). These results are alsoconsistent with one-color 2D experiments by Nemeth et al. (2008) whoobserved modulation of the central peak (Nemeth et al., 2008).

A common problem facing studies of vibrational wavepacket motion isthat it is difficult to determine whether the observed vibrational motionoccurs on the ground or the excited electronic state (Armstrong et al.,2003; Kobayashi et al., 2001; Kumar et al., 2001). This distinction is often ofinterest for identifying vibrational modes involved in a chemical reaction.2DES separates ESA contributions which are opposite in sign and oftendisplaced relative to the other signal components as in Figure 6. Modula-tion of the amplitude of ESA features of the data must arise solely fromvibrational wavepacket motion on the excited state, allowing for a clearseparation from ground state contributions. This has not yet beenobserved in 2DES data.

2.3.3 Understanding 2DES Spectra

Descriptions of the quantum dynamics of condensed phase systemsremain an unmet challenge. A powerful and extensively used formalismfor describing nonlinear spectroscopies is based on the nonlinear response



function. In general, the response of a medium to the application of thethree time-ordered excitation pulses is given by the third-order responsefunction R3D (Mukamel, 1995), the frequency-domain analog of which isthe third-order nonlinear susceptibility w 3ð Þ (Butcher & Cotter, 2003).In the impulsive limit, the radiated field emitted in response to theexcitation pulse sequence is directly proportional to R3D. The problemthen becomes one of using the measured R3D to derive a picture of thestructural and electronic dynamics of the system. Figure 18 depicts aflowchart of the approach that is often used to obtain an accurate systemdescription to reproduce the amplitudes, positions, and lineshapesrevealed by the 2DES spectra (Khalil et al., 2003).

The starting point is a set of generalized coordinatesQ that provide anappropriate description of the system. The system properties, such as thesite energies of the constituent chromophores, and couplings betweenchromophores, are described by the Hamiltonian ~H

e

S. In the case oflight-harvesting complexes, and other molecular aggregates, the Frenkelexciton Hamiltonian has been extensively used (Davydov, 1971). Thedegree of coupling between chromophores determines what theory isappropriate to describe the coupling: in the weak-coupling limit Forstertheory is adequate, while intermediate to strong coupling requires alter-native descriptions such as modified Forster theory, which accounts forthe finite spatial extent of the chromophores ( Jordanides et al., 2001;Scholes, 2003; Scholes & Fleming, 2000; Scholes et al., 2001). Diagonalizing

3D responsefunction: R3D

2DES spectra

Molecular structure and dynamics: Q

System eigenstates:HS

Local coordinatesand coupling: HS

e

System bathinteraction: HSB

Bath: HB

FIGURE 18 Flowchart for using 2DES to derive an accurate description of the

structural and electronic dynamics of our system. Modeled after Khalil et al. (2003a,b)



the ~He

S Hamiltonian results in the electronic eigenstates whose transitionfrequencies and lineshapes map onto the peak positions in a 2DES spec-trum at t2 ¼ 0.

In the condensed phase the system is constantly interacting with thesurrounding bath, causing fluctuations and shifts in transition energies ofthe eigenstates. The nature of the bath is described by the bath Hamilto-nian HB, while its interaction with the system is captured by the system–bath coupling HSB. Because many of the bath degrees of freedom mayhave little effect on the system, one popular approach to simplifyingquantum condensed phase dynamics is based on the reduced densitymatrix formalism, in which degrees of freedom of the bath that areconsidered to be irrelevant to the measured observable and are averagedover. Redfield theory, one famous implementation of this procedure(Redfield, 1965), reduces to the Bloch model in the case of a two-levelsystem. Originally developed for NMR spectroscopy where interactionsbetween the system and bath are weak, Redfield theory fails in manycondensed phase systems. For example, the solvation of a dye molecule,where the excited state induces reorganization of the surrounding solventstructure, requires a description that captures the effect of the system onthe bath coordinates. The secular approximation, which is often used tofurther simplify Redfield theory, uncouples the diagonal and off-diagonalelements of the density matrix and therefore cannot describe effects suchas coherence transfer. Seeking alternate descriptions to condensed phasequantum dynamics is an ongoing effort (Brixner et al., 2005; Cho et al.,2005; Zhang et al., 1998; Zigmantas et al., 2006) that can be refinedwith theaid of 2D spectroscopy: for any given system description, the third-orderresponse function can be directly computed, allowing calculation of theresulting 2DES spectra and an iterative approach to improving the systemdescription to match 2D experiments.

3. TWO-DIMENSIONAL VIBRATIONAL SPECTROSCOPY

In his first report of two-dimensional NMR spectroscopy, Richard Ernstrecognized that the method was general and in principle could be appliedto other spectral regions such as ‘‘electron spin resonance, nuclear quad-rupole resonance, in microwave rotational spectroscopy, and possibly inlaser infrared spectroscopy’’ (Aue et al., 1976). Although it took more thantwo decades to develop methods of generating ultrashort, broadband IRpulses, and efficient electric field detection techniques, today there aremany groups using 2DIR spectroscopy to study a diverse array of molec-ular systems. This section outlines idiosyncrasies of multidimensionalspectroscopy in the infrared, then summarizes the experimental



implementation of 2DIR spectroscopy, and considers several illustrationsof both equilibrium and nonequilibrium variations of the technique.

3.1 Idiosyncrasies of Multidimensional IR Spectroscopy

In the infrared, transitions generally correspond to relatively well-localizedvibrational excitations that are often associated with specific chemicalgroups. The trend one expects in electronic transitions—that transitionsbetween states that are extended in space are lower in energy, while thosethat are localized are higher in energy—are sometimes, but not alwaysmaintained (Harris & Bertolucci, 1989). For instance, the O–H stretch,which appears between 3300 and 3500 cm�1, is highly localized when theOH unit is attached to something very different, such as the CH3 (methyl)group in methanol, CH3OH. Likewise, the often studied ‘‘amide I’’ band ofproteins and peptides, is largely composed of the C¼O stretch and appearsaround 1650 cm�1, though there is some deformation along the C–N bond.When themolecule has high symmetry, however, the coupling between thelocal units often leads to delocalized vibrational modes that span the entiremolecule, even at very high frequencies. Examples include the�2000 cm�1

CO stretch band in metal carbonyl complexes such as Mn2(CO)10 and the�3300 cm�1 CH stretches in benzene.

A key feature of vibrational spectroscopy is that it is possible to shift theenergy of a transition by isotopic substitution. For example, the C¼O in theamide I band of a helical peptide absorbs at roughly 1650 cm�1, and thisband shifts to the red by 60 cm�1 upon substitution to 13C¼18O (Backuset al., 2008b; Bagchi et al., 2007; Bredenbeck & Hamm, 2003; Fang &Hochstrasser, 2005; Fang et al., 2003, 2004, 2006; Ganim et al., 2008;Ihalainen et al., 2008; Kim & Hochstrasser, 2005b; Kim et al., 2005, 2008;Mukherjee et al., 2004, 2006a,b; Pfister et al., 2008; Smith & Tokmakoff,2007a,b; Wang et al., 2006; Woutersen & Hamm, 2001). Using site specificisotope labels, it is possible to selectively investigate the coupling between aparticular pair of amide units. Applying site specific isotope labels topeptides (small chains of amino acids) is straightforward using directpeptide synthesis, but extending the approach to whole proteins is notcurrently feasible since living systems—bacteria or yeast—are typicallyused to produce complete proteins. Besides the technical challenges ofisotope editing, there are more substantial consequences, particularlywhen multiple transitions on distinct sites are coupled either through themechanical anharmonicity or through transition dipole coupling. Shiftingone unit out of resonance with the others introduces a defect and qualita-tively changes the resulting eigenstate and its associated degree of delocali-zation. Eigenstate maps have been constructed for large protein systemsshowing the effect of isotope editing on distant sites (Smith & Tokmakoff,2007a). In the metal carbonyl dimanganese decacarbonyl (Mn2(CO)10, or



DMDC), the 1% natural abundance 13C amounts to 10% of the samplecontaining an isotope defect. There are two different CO positions on themolecule, resulting in twodifferent sets of vibrationsdue to the isotopomers.Using quantum chemistry theory, it is possible to evaluate the fundamentalvibrational eigenstate energies for both the pure and isotopomer cases.Thus, isotope editing can be very useful in providing additional spatialinformation, but the effects are not always local and need to be treatedwith care.

3.2 Experimental Implementation

Most 2DIR spectrometers employ 1-kHz amplified Ti:sapphire lasers thatproduce 1–5 mJ pulses with pulse durations <100 fs centered at a wave-length of 800 nm. To generate mid-IR pulses, the 800 nm acts as a pump ina collinear optical parametric amplifier (OPA) seeded by a white-lightcontinuum generated in a Kerr material such as sapphire. Several nonlin-ear optical materials are available for the parametric amplification, withspecific choices based on the desired spectral range. Following two-stageparametric amplification, the signal and idler are difference frequencymixed in another crystal, producing output in the mid-IR (Kaindl et al.,2000). Our implementation (Figure 19) uses a single 4-mm b-bariumborate (BBO) crystal for both the first and second stages of the OPA,followed by a 1-mm GaSe crystal for the difference frequency generation.Using beam splitters in the OPA we arrange a second identical OPAalongside the first, permitting a second independently tunable OPAwhich provides another pair of signal and idler outputs. The dual OPA/DFG provides the capability of exciting and detecting at two differentspectral regions to study coupling between different kinds of vibrations(Kumar et al., 2006; Rubtsov et al., 2003a,b, 2005). The first two pulses in thenoncollinear three-pulse photon echo sequence are derived in an interfer-ometer using a single CaF2 beamsplitter and the delays are scanned usingtwo pairs of 7.3� apex, 1 inch ZnSe wedges. The wedges are preferred overretroreflectors because of the greatly reduced periodic artifacts induced byslight mirrormisalignments (Ding et al., 2006). Thewedges aremounted on1-inch translation stages actuated by optically encoded (7.4 nm resolution)DC motors that are scanned once, giving the t1 axis. The large apex angleoffers a maximum delay of 12 ps, corresponding to a Fourier transformresolution of 2.4 cm�1, which is sufficient for themajority of IR transitions inthe condensed phase. Other implementations of 2DIR spectroscopy employthe pulse-shaping and partially collinear methods described in Section 2.

The features of the above description are common to most implemen-tations of background-free Fourier transform 2DIR spectroscopy. One keypractical advantage of visible 2D spectroscopy over the IR counterpart isthe ease of single-shot spectral measurements using a silicon CCD camera

Ti: sapphire

CCD

LiNbO3

Spec

Reference

Sample

Signal

DFG

k3

k2

k1

DFG

ZnSe

ZnSe

800 nm chirped

800 nm shortpassD

ual-O

PA

FIGURE 19 Experimental apparatus for chirped-pulse upconversion detected 2DIR

spectroscopy. The two OPA outputs generate two independently tunable 100-fs mid-IR

pulses. The time delays between the first two pulses are achieved using movable ZnSe

wedges. Sum frequency generation between the chirped 800-nm pulse and the IR

field (signal and local oscillator) takes place in a 0.2–0.8-mm wedged 5% doped MgO:

LiNbO3 crystal. The visible signal is recorded using a CCD camera in single-shot mode

at 1 kHz repetition rate



in a spectrometer. In contrast to megapixel cameras for visible lightdetection, the state of the art of IR array detectors based on mercury–cadmium–telluride (MCT) used to measure the multichannel IR spectrumconsist of merely 128 pixels. Moreover, these expensive detectors are notuseful outside of the IR spectral range, limiting use of the spectrometer toone purpose. In a series of demonstrations (Baiz et al., 2008; Kubarychet al., 2005; Lee et al., 2008; Nee et al., 2007, 2008; Treuffet et al., 2007; Annaet al., 2009), we have shown that upconverting the IR signal intothe visible—where a silicon CCD camera can detect the spectrum—is afeasible and flexible alternative to direct IR detection.

Although it is not strictly required, we have adopted a strategy wherethe IR signal is mixed with a highly chirped 800-nm pulse that is derivedfrom the uncompressed output of the Ti:sapphire chirped-pulse amplifiersystem. Chirped pulse powers of 500 mJ are easily produced with modestloss of the amplified output for our 2.5-mJ Spectra-Physics Spitfire Prosystem. The linear frequency slope due to the predominantly second-order spectral phase introduced by the stretcher was measured usingfrequency-resolved cross-correlation with a temporally short IR pulse to



be 0.5 cm�1 ps�1 (0.015 THz ps�1, or 66.7 ps2, or 6.67�107 fs2). Given thatthe full, usable, base-to-base bandwidth of the chirped pulse is 200 cm�1,the base-to-base temporal width is 400 ps assuming a purely linearfrequency chirp. This base-to-base width corresponds to a Gaussian full-width-at-half maximum of 165 ps. Since the emitted nonlinear signal has afinite temporal extent, there is some inevitable broadening of the spectralresolution due to the overlap with a finite spectral band of the chirpedpulse. For a 5-ps exponential decay, which corresponds to a 1.33 cm�1

spectral FWHM, the signal overlaps with 2.5 cm�1 of chirped pulse band-width, leading to some degree of spectral convolution. Since the chirp ofthe mixing pulse is known, however, there is a predictable effect on theupconverted field provided one knows the spectral phase of the chirpedpulse. In principle, the spectral distortion associated with the chirpedpulse upconversion process can be corrected in a simple way (Annaet al., unpublished).

In practice, the chirped pulse energy is roughly 100 mJ and the upcon-version crystal is an uncoated 0.2–0.8 mm MgO(5%):LiNbO3 wedge. Thewedge is required to avoid temporal modulations of the chirped pulse asit interferes with itself after two reflections. Due to the very long coher-ence length of the chirped pulse, even the relatively small Fresnel reflec-tivity of the uncoated crystal faces induces temporal modulations of thechirped pulse amplitude. These modulations have a period on the orderof 5–10 ps for typical crystal lengths (0.2–0.5 mm), leading to both tempo-ral and spectral modulations of the upconverted fields. The wedgedcrystal both rejects the reflected signal and acts as a variable thicknesscrystal, enabling an effectively continuously adjustable analog gain of theupconverted signal. There is always a trade-off between phase-matchingbandwidth and upconversion efficiency, and the wedge allows a latitudeof choice for a specific spectroscopic context. Since we implement upcon-version using a noncollinear geometry to avoid a dichroic recombiningoptic, we arrange the wedge of the crystal to be oriented perpendicular tothe phase-matching axis, thus the prismatic effect of the wedge partiallycompensates the spatial chirp of the noncollinear phase-matching.

Besides the Fourier-transform-based 2D techniques described above, itis also possible to record absorptive 2D spectra using a transient hole-burning implementation, and indeed this was the method first used adecade ago (Hamm et al., 1998). A short path length cavity, low-finessescanning Fabry–Perot interferometer produces an exponentially decayingpulse in the time domain, which has a Lorentzian spectral profile.By tuning the cavity length, the center frequency of the etalon shifts, andat each center frequency the differential absorption spectrum (pump-onminus pump-off) of a broad-band probe is measured. Compiling the dif-ferential absorption spectra results in an absorptive 2D spectrum withoutany need for phasing the data. The pump-probe spectrum contains a signal



with a perfectly defined phase relationship with the local oscillator, whichis the unabsorbed probe itself. The interference of the reference and theemitted third-order signal produces a purely absorptive 2D spectrum.A limitation of the Fabry–Perot filter approach is the temporal profile ofthe pump pulse, which is peaked at early times and decays at later times.To ensure well-separated pulses, the probe pulse cannot enter the samplebefore the pumppulse has had time to decay appreciably. Thus the spectralresolution of the o1 axis is linked to the time resolution during t2 (Cervettoet al., 2004). In practice, the probe is generally delayed roughly 1 ps relativeto the pump, allowing the system to vibrationally relax or for the initiallyexcited IR excitation to redistribute among any modes that energeticallyand spatially overlap the initial excitation. A modification of the temporalprofile was implemented using the pulse-shaping approach producing atime-reversed exponential pulse, enabling a closer temporal spacingbetween pump and probe (Shim et al., 2007). Despite the technical limita-tion of the hole-burning approach, some of the most definitive and seminal2DIR spectra, particularly of nonequilibrium systems have been measuredusing this approach (Hamm et al., 2008).

3.3 Examples of Equilibrium 2DIR Spectroscopy

There have been several reviews summarizingmany of the achievements of2D spectroscopy (Ganim et al., 2008; Hamm et al., 2008; Jonas, 2003; Khalil,2003a; Wright, 2002; Zheng et al., 2007); particularly noteworthy is a recentcomprehensive overview by Cho (2008). Rather than attempt to list all thevaried systems that have been studied,we aimhere to show a few selectionsthat illustrate some common molecular goals outlined in Section 1.4.

3.3.1 The OH Stretch in Water

The infrared spectra of all hydrogen-bonded molecules—most notablywater—are characterized by a signature broad absorption featurebetween 3000 and 3700 cm�1 corresponding to either the OH or NHstretch. This broad feature is much-maligned by chemists due to its lackof specificity, but inside this complex band hides much of the richness ofthe unusual structural and dynamical properties of liquid water.Although the OH stretch in the gas phase is a very narrow feature, inthe liquid it is significantly broadened to roughly 300 cm�1 due to thestrong hydrogen bonding interactions. From diffuse X-ray scattering,water is known to have short range order, where, on average, eachwater molecule donates and accepts two hydrogen bonds, leading to afour-coordinate structure (Hura et al., 2000; Sorenson et al., 2000). It hasbeen a long-standing challenge in the study of water dynamics to under-stand the timescale of the structural rearrangements of liquid water (Reyet al., 2002). From its remarkable spectrum, it is clear that water presents a

1.0

0.8 OO

O

HH

HH

H HHH

O

0.6

0.4

OH

str

etch

abs

orba

nce

(a.u

.)

0.2

0.03000 3200 3400

Wavenumber (cm−1)

3600 3800

FIGURE 20 Schematic OH stretch spectrum of liquid water. The spectrum is nearly a

perfect Gaussian, where the low-frequency region corresponds to short hydrogen

bonds and the high-frequency side is attributed to long hydrogen bonds. 2DIR

experiments show that the ‘‘homogeneous’’ line width varies with position in the band



spectacular example of inhomogeneous broadening, and this inhomoge-neity cannot be static since the liquid is constantly fluctuating. As dis-cussed above in general terms, 2D spectroscopy is particularly well suitedto the task of measuring the loss of transition frequency memory due tospectral diffusion. Thus, the development of 2DIR spectroscopy has to acertain extent been guided by the quest to dissect the complex lineshapeof water.

The basic description of the water OH stretching band is linked to arather simple geometrical picture (Figure 20) where the dominant param-eter is the distance between the two oxygen atoms involved in the hydro-gen bond. In a ‘‘long’’ hydrogen bond, the O atoms are farther apart thanthe average, leading to a higher OH stretch frequency as it shifts towardthe gas phase value where the hydrogen bonds are absent. Conversely, a‘‘short’’ hydrogen bond, where the O atoms are closer than the average,has a lower OH stretch frequency due to the partial sharing of the H atombetween the two O atoms. Although the distance between oxygen atomsis the dominant determinant of OH frequency, recent studies have foundthat more detailed, multidimensional coordinate spaces are useful fordescribing hydrogen bond switching, where the motion of the H atom issignificantly far from the equilibrium potential minimum (Eaves et al.,2005a; Loparo et al., 2006b). The OH stretch in water illustrates a proto-typical case where there is a well-defined structure-spectrum relation-ship, but the underlying dynamics are entirely hidden beneath thestructureless lineshape.



Pure water at room temperature and atmospheric pressure has aconcentration of 55.6 mol l�1 (33.4 molecules nm�3), making the concen-tration of OH stretch vibrations 111.2 mol l�1. Faced with the difficulty ofassembling a practical sample cell of pure water, coupled with the analyt-ical complexity of a liquid composed entirely of the same spectroscopicentity, the first ultrafast 2D measurements were performed on the relatedisotopomers, HOD in a solution of H2O (HOD:H2O) (Asbury et al., 2004a,b,c) and HOD in a solution of D2O (HOD:D2O) (Eaves et al., 2005a,b;Loparo et al., 2006a,b). Using dilute isotope substitution, the spectroscopicmode—either OD in (HOD:H2O) or OH in (HOD:D2O)—is shifted fromthe H2O absorption, lowering the optical density and permitting the useof typical 10–100 mm path length samples. Besides the practical advan-tages of using the isotope substituted water systems, they also permit thestudy of a single OH or OD stretch, effectively in spectroscopic isolation,as a probe of the surrounding liquid dynamics.

The most striking feature of the isotopically substituted water systemsis shown schematically in Figure 20 as the underlying lineshapes. From themeasured 2DIR spectra, both Fayer and coworkers and Tokmakoff andcoworkers observed that the blue side of the band had a broader linewidththan the red side of the band. These line widths are obtained by consider-ing antidiagonal slices as depicted in Figure 3. The appearance of the blue-broadened lineshape is surprising if one imagines that in the limit of aninfinitely long hydrogen bond one recovers the gas phase picture. Whilethe frequency is shifted toward the blue as expected, the line width isinconsistent with the narrow gas phase width. On the red side, the linewidth is actually narrower despite greater interactions with the surround-ings due to the shorter hydrogen bond. These heterogeneous lineshapesindicate that the longer hydrogen bonds have faster short-time dynamicsdue to the orientational freedom gained by the decreased interaction withsurrounding molecules (Kwak et al., 2008a,b). Conversely, the shorterhydrogen bonds are more rigid since they are essentially held in place bythe stronger association with the adjacent hydrogen bond acceptor(Loparo et al., 2006a,b; Steinel et al., 2004). In HOD:H2O, the asymmetrydecay due to spectral diffusion occurs on a timescale of roughly 3 ps(Asbury et al., 2004a,b,c; Corcelli et al., 2004), whereas in HOD:D2O, therelaxation is roughly twice as rapid (Loparo et al., 2006a,b). Thesehydrogen-bond rearrangement dynamics reflect the well-known time-scale for solution phase phenomena such as solvation dynamics (Castner& Maroncelli, 1998; Stratt & Maroncelli, 1996).

Inspired by the experimental results, theoretical investigations havepointed to the importance of another aspect of condensed phase systemsthat has no obvious isolated-atom analogue, namely the independence ofthe transition dipole moment on the solvent coordinate. The so-calledCondon approximation (Condon, 1926) treats the transition dipole as an



operator solely in the system’s Hilbert space and neglects the influence ofbath degrees of freedom. Because coupling to the bath is often weak—soweak that it is almost always treated perturbatively—it is generally rea-sonable to assume that the solvent does not strongly affect the system’scoupling to the electric field. Nevertheless, using models that account fornon-Condon effects to varying degrees, it was shown that a modelemploying the pure Condon approximation was in poorer agreementwith the data than one which does not invoke the approximation at all(Schmidt et al., 2005, 2007). Although non-Condon effects are manifestedin the linear spectrum, the added frequency axis and temporal dynamicsof nonlinear spectroscopy enable more highly constrained feedback to thetheoretical analysis. Non-Condon effects have also been predicted theo-retically for a model of proton transfer in solution (Hanna & Geva, 2008a),where the transition state geometry actually enjoys a much larger transi-tion moment—in effect magnifying the transition state. It is a holy grail ofultrafast spectroscopy to be able to observe transition states, and suchfortuitous enhancement of the transition state region due to non-Condoneffects may aid in achieving this long-standing goal.

The spectroscopy of pure water is not the same as what one observes inthe isotope substituted systems for the simple reason that the line between‘‘system’’ and ‘‘bath’’ is very hard to draw since the two are essentiallyfully resonant. The resonant interaction between the optically excitedmode and its surroundings introduces two significant complications:(1) energy transfer can become qualitatively more efficient due to reso-nant coupling (Woutersen & Bakker, 1999) and (2) the spectroscopic modeis less well defined due to the spatial delocalization of the energy eigen-states (Paarmann et al., 2008). Forster coupling between nearby oscillatorsis virtually assured, as well as an efficient energy relaxation processthat passes through the second overtone of the HOH bending vibration(o ¼ 1650 cm�1) (Ashihara et al., 2007; Lindner et al., 2006). Notwith-standing the technical difficulties of studying an ultra-high optical den-sity sample, the pure water system posed a daunting challenge.

Using an ingenious nanofabricated SiC flowing sample cell with a400 nm pathlength, combined with the passive phase stability of adiffractive optics-based 2D spectrometer, Miller and coworkers suc-ceeded in recording the 2DIR spectrum of pure water (Cowan et al.,2005; Kraemer et al., 2008). While they did observe a heterogeneouslineshape with a broader blue side, they also observed a dramaticallyincreased 50-fs decay of the inhomogeneous asymmetry. Such a rapidloss of frequency memory could not simply be due to structural rear-rangements of the hydrogen bonded network, since it is the samenetwork in both pure water and the isotope substitutes—only thespectroscopic probe has changed. The most current interpretation,based on temperature-dependent measurements as well as mixed



quantum-classical simulations, points to eigenstate delocalization asthe principal mechanism for the loss of frequency memory, as well asrapid anharmonic coupling to lower frequency intermolecular modes(Ashihara et al., 2007; Paarmann et al., 2008). Due to the strong degreeof resonant coupling the instantaneous eigenstates span multiple OHchromophores, much in analogy to the multichromophoric systemsinvolved in natural and artificial light harvesting (Cho et al., 2005;Cheng et al., 2007; Cheng & Fleming, 2008; Scholes & Fleming, 2000;Scholes et al., 2001; Zigmantas et al., 2006). As the liquid fluctuates to adegree that might be imperceptible using the local probe of an isotopesubstitute, the composition of the spectroscopically addressed statechanges spatially and energetically. Thus, there are effectively twoclasses of spectral diffusion, one at the longer range length scale ofthe hydrogen bond network, and another at the shorter range of eigen-state delocalization and anharmonic coupling. The eigenstates areviewed as fragile markers of a much more detailed set of fluctuationsrelative to the coarse hydrogen bonding rearrangements probed inHOD:H2O and HOD:D2O. Liquid water illustrates the powerful cap-abilities of multidimensional spectroscopy to dissect a complex spectrallineshape that has stood for years as one of the most important puzzlesin condensed phase spectroscopy.

3.3.2 Vibrational Coherence

Uncovering the heterogeneous dynamics of water illustrates a fundamen-tal strength of 2D spectroscopy to elucidate underlying dynamics thatmake up a spectral lineshape. The 2D water experiments reported to date,however, focus specifically on one single band and do not reflect directlythe coupling of multiple spectral features. Besides the lineshape, 2Dspectroscopy is designed to measure couplings through the observationof cross-peaks in the spectrum. One aspect of coupling is the possibility ofobserving coherent modulation of cross-peak amplitudes as a function ofthe waiting time t2. The observation of coherence in ultrafast spectroscopyis by no means new (Elsaesser & Kaiser, 1991), but multidimensionalspectroscopy provides a clear window to observe the individual statesinvolved, as well as to determine their precise dephasing dynamics.Considering for simplicity a three-state system (Figure 7), composedof a common ground state and two optically accessible excited states,following the first two field interactions the possible system densities arer00, raa, rbb, rab or rba. For the rephasing sequence, the t2 coherencesappear at the locations of the cross-peaks since the coherence during t1is at one frequency, r0a or r0b, and the coherence during t3 (bold arrow) isat the other frequency, r0b or r0a, respectively (see Figure 7). For the



nonrephasing sequence, both pathways that go through a coherenceduring t2 involve the same t1 and t3 coherences, thus, the t2-dependentcoherences appear on the diagonal.

Since the system may evolve in a coherence during the waiting time,the signal will oscillate at the difference frequency of the two constituentlevels. The modulation can be easily verified by considering both thecoherent and ground state paths. The coherent paths are represented bythe following density matrix:

rðcoherentÞ0 0 00 0 rab0 rba 0

0@

1A: (8)

A full derivation of the coherence appearing in cross-peaks that arise froma common ground state is given in the Appendix. We find the total signalat the cross-peak (o1 ¼ o0b, o3 ¼ o0a) to be given by the sum of thecoherence path and the ground state path:

S totalð Þ ¼ S coherenceð Þ þ S groundð Þ ¼ e�ioab t2�t1ð Þ þ 1h i

jm0aj2jm0bj2~r00; (9)

where ~r00 is the initial ground state density (which could be a thermaldistribution, for example), m0a and m0b are the transition dipole momentsbetween j0i and jai and jbi, respectively. The two constants are identicaland are given by the square magnitudes of the individual transitiondipole moments between the ground state and the two states involvedin the coherence. At this point it is straightforward to include the dephas-ing of the coherence through a phenomenological damping factor, g.Including the dephasing yields:

S totalð Þ ¼ e�i oabþigð Þ t2�t1ð Þ þ 1h i

jm0aj2jm0bj2~r00: (10)

A characteristic curve of the coherence cross-peak is shown in Figure 21for parameters typical of metal carbonyl complexes. This feature isobserved experimentally and will be described below.

It is possible to observe t2-coherences in several ways using multidi-mensional spectroscopy. In an absolute-value rephasing spectrum, thecross-peaks are modulated according to the discussion above, whereasin an absolute-value nonrephasing spectrum the diagonal peaks aremodulated. In a purely absorptive spectrum, where both a rephasingand nonrephasing spectrum are measured and their real parts added,both the diagonal and cross-peaks oscillate. Depending on the spectral

2.0

1.5

1.0C

ross

pea

k m

agni

tude

(a.

u.)

0.5

0.00 2 4

t (ps)6 8 10

FIGURE 21 Simulated cross-peak decay due to introducing a dephasing of the

coherence with a frequency of 62 cm�1 and a dephasing rate of 0.5 ps�1



congestion, and the added difficulty of having to record many 2D spectrato track the full timescale of the coherence, absolute-value rephasingspectra have proved capable of resolving complex coherent dynamics(Nee et al., 2008).

Since third-order multidimensional spectroscopy is a four-wave mix-ing process, it is possible to view the signal generation as a phase andamplitude grating written in the material (Eichler et al., 1986). In atransient-grating experiment, two temporally overlapped pulses arriveat a sample and are tuned to match a resonance in the material. The anglebetween the beams leads to a spatial interference pattern that encodesexcited state and ground state molecules in the bright and dark fringes,respectively. The diffraction efficiency of a time-delayed probe pulsedecreases as the molecules return to the ground state, and the diffractioncan be due to a combination of refractive and absorptive changes viaa Kramers–Kronig relation (Dadusc et al., 1998; Ogilvie et al., 2002b).To measure separately either the refractive or absorptive components ofthe signal, one requires a phase-sensitive method based on interfering thesignal with a local oscillator reference, as has already been describedabove for 2D spectroscopy (Dadusc et al., 2001). From the perspective ofa transient-grating experiment, the importance of the ground state path-ways is evident: the signal relies equally on the presence of molecules inan excited state population as it does on those that remain in a groundstate population.

The grating picture also helps to clarify how to imagine t2-coherencesfrom a macroscopic perspective. Since the coherence necessarily involvestwo different excited states, the grating is written with two fields ofdifferent frequency, generating a traveling wave. As the waiting time



between writing and readout is increased, the diffraction pattern due tothe coherence moves across the sample at a rate proportional to thedifference frequency, leading alternately to overall constructive anddestructive interference with the static population grating (Figure 22).Although the simple grating experiment performed with broadbandpulses does not permit correlation of an excitation and detection fre-quency, the generation of coherences can still be viewed within themacroscopic and physically intuitive context of a transient diffractiongrating.

The experimental and theoretically predicted linear infrared (FTIR)spectrum of the metal carbonyl complex Mn2(CO)10 (dimanganese dec-acarbonyl, DMDC) is shown in Figure 23 along with the molecule’sstructure. The largest features are the three bands centered at 1982, 2013,and 2045 cm�1. The central peak is due to two degenerate modes of Esymmetry, thus the feature is roughly twice the magnitude of the othertwo. Based on the molecule’s symmetry, one predicts and observes fourIR active vibrations, leaving six Raman active modes that are invisible inthe IR (Cotton & Kraihanzel, 1962). That there are more dark modes thanbright modes immediately points to a key complication in a system suchas DMDC since the anharmonic potential is unrelated to the spectroscopicactivity, indicating that vibrational energy is free to flow between brightand dark modes. A full accounting of the relaxation processes in DMDCrequires the inclusion of the dark modes, though they are not directlyobserved in the 2DIR experiment. In the experimental spectrum there aresmall-amplitude peaks in addition to the three main bands, and these areassigned to the presence of 13C. The most clearly visible feature due to 13Cin the FTIR spectrum shown in Figure 23 is that at 2002 cm�1. The naturalabundance of 13C is 1.1%, indicating that 10% of the sample isMn2(CO)9

13CO. Since the modes are delocalized throughout themolecule,one 13CO will have an effect on essentially all of the vibrational modes,changing the linear and nonlinear spectrum in a generally unintuitivemanner. There are two structurally distinct carbonyl locations in DMDC,8 are equatorial and 2 are axial. Assuming statistical independence of theisotope substitution site, 8% of the sample has an equatorial isotopedefect, while 2% has an axial defect. Since the heterodyne detected 2DIR

Δx

FIGURE 22 Illustration of the traveling complex grating created by the launching of a

coherence. As the traveling grating sweeps across the static grating due to the

noncoherence pathways, the signal at the cross-peak oscillates

1940

19481950

1956

1983

2002

2044

2014

1980

1993

2036

2005

1960

20x

1980

FTIR in n-hexane

DFT (BP86)A

bsor

banc

e (a

.u.)

2000

Frequency (cm−1)

2020 2040 2060 2080

FIGURE 23 Experimental (blue) and theoretical (green) linear IR absorption spectrum

of Mn2(CO)10. The experimental result is for a 2 mM solution in cyclohexane, whereas

the theoretical result added a broadening of 5 cm�1. The structure of the molecule is

also shown



signal has linear scaling with number density, contributions from speciespresent in these ratios are certainly observable. The isotope contributionadds some complexity to the spectrum, and in principle can be removedeither by chemical enrichment with 12CO or by direct synthesis of anisotopically pure sample. Alternatively, it has long been a technique incharacterizing metal carbonyl compounds to purposefully introduce13CO to break the symmetry, thus relaxing the selection rules.

The absolute-value rephasing 2DIR spectrum of DMDC is shown inFigure 24 for a waiting time of �100 fs. As is expected from a three-peakedlinear spectrum, the 2D spectrum has ninemajor peaks corresponding to allof the excitation/detection pairs. Besides these nine peaks, which corre-spond to transitions between the ground state and the first excitedmanifold,there are additional peaks that arise from ESA. The ESA peaks reflectdirectly the frequencies of the second excited manifold, and are thereforedue to the diagonal and off-diagonal anharmonicity initially introduced inSection 1.4. From all of these peak positions and their relativemagnitudes, it

2060

2040

2020

2000w3/

2pc

(cm

−1)

w1/2pc (cm−1)

1980

2060

14

x 105

12

10

8

6

4

2

20402020200019801960

1960

FIGURE 24 Experimental absolute magnitude rephasing 2DIR spectrum of 3mM

Mn2(CO)10 in hexane solution. Red dots show the locations of the calculated vibrational

anharmonicities using ab initio quantum chemistry



is possible to devise a Hamiltonian that is consistent with the data, but thenumber of fitting parameters is so large that it would be difficult to find aunique solution. The approach we have taken is to compute the harmonicvibrational frequencies using ab initio quantum chemistry, and then usevibrational perturbation theory to compute the vibrational anharmonicities.Particularly whenwewish to include all 10 carbonyl modes (6 of which areRaman active), diagonalization of a 10-dimensional problem is intractablegiven that each dimension requires on the order of 10 basis vectors to reachconvergence. In general if one requires nB basis vectors for each of Ddimensions, the number of total vectors in the tensor product space is nB

D,and the Hamiltonian matrix has dimension nB

2D. Thus, for a 10 modesystem with 10 basis vectors per dimension, the Hamiltonian contains 1020

elements, or 400 exabytes of memory for single precision.The 2D spectrum shown in Figure 24 was recorded by one single scan

of the t1 delay, which required only 10 s of acquisition time. This rapidmeasurement enables a fine sampling along the t2 axis, recording aseparate 2D spectrum at each waiting time point. Figure 25 shows four2D spectra taken at various waiting times, showing that the cross-peaks

2040

t2= 0 ps t2= 0.3 ps t2= 0.6 ps t2= 0.9 ps

2020

2000

2000

−w1/2pc (cm−1) −w1/2pc (cm−1) −w1/2pc (cm−1) −w1/2pc (cm−1)

w3/2p

c (c

m−1

)

2000 20002020 2020 20202040 2040 20401980 1980 2000 2020 20401980

1980

0 0 20 40 60 80 100

FT

inte

nsity

(a.

u.)

0 20 40 60 80 100

0 20 40 60 80 100

4

(a) (b)

(d)

(f)

(c)

(e)

8 12 16

0Pea

k am

plitu

de (

a.u.

)

4 8 12 16

0 4Waiting time (ps) Oscillation wavenumber (cm−1)

8 12 16

1980

FIGURE 25 (Top) A series of absolutemagnitude rephasing 2DIR spectra of Mn2(CO)10in cyclohexane solution at different values of the waiting time t2. The cross-peaks are

modulated due to the creation of coherences among the excited eigenstates. (Bottom)

A series of t2-traces and their Fourier transforms at various cross-peak locations (o1,o3)

in cm�1: (a, b) (black) (2013, 1983) (red) (1983, 2013); (c, d) (black) (2045, 1983) (red)

(1983, 2045); (e, f) (black) (2045, 2013) (red) (2013, 2045)



exhibit modulations as a function of t2. These modulations are due to thecoherence pathways described above in the context of a three-state model.The t2-dependence can be seen more clearly by following individualcross-peaks in the t2 domain. Figure 25 also shows the modulation ofthree cross-peaks taken from 180 2DIR spectra recorded from t2 ¼ 0 to18 ps in 100-fs steps (Nee et al., 2008). It is noteworthy that the temporalprofile of the modulations exhibits cusps near zero-amplitude andsmoother changes near the maximum amplitude, in agreement with thederivation above. Fourier transforms of the coherence signals with respectto t2 are also shown. The highest frequency coherence centered near



62 cm�1 (2045–1983 cm�1) has a spectral width that is very narrow,whereas the other coherences are broader. The broadening of the lower-frequency coherence is likely due to spectral overlap with other transi-tions, possibly from the isotopomers. Based on the most cleanly resolved62-cm�1 feature, the dephasing time of the coherence is comparable tothat of the fundamental transitions as determined be either the line widthof the 1D FTIR spectrum (assuming homogeneous broadening) or fromthe antidiagonal width of the 2D spectrum.

Since the frequency of the coherence can be straightforwardly pre-dicted from the linear spectrum, the key new information obtained inthe t2-dependent measurement is the line width (or dephasing rate) of thecoherence. It is possible to derive a relationship between the line width ofthe ‘‘excited state’’ coherence, rab, in relation to the line width of the‘‘fundamental’’ coherence r0a and r0b. The instantaneous energy gapbetween the states jai and jbi and the ground state j0i is (in terms offrequency) (Nee et al., 2008):

o0a tð Þ ¼ ho0ai þ do0a tð Þ; o0b tð Þ ¼ ho0bi þ do0b tð Þ; (11)

where the angled brackets denote the average transition frequency, and dois the time-dependent fluctuation. The frequency–frequency correlationfunctions that determine the ‘‘fundamental’’ coherence line widths are:

c0a tð Þ ¼ ho0a tð Þo0a 0ð Þi ¼ hdo0a tð Þdo0a 0ð Þi;c0b tð Þ ¼ ho0b tð Þo0b 0ð Þi ¼ hdo0b tð Þdo0b 0ð Þi; (12)

With the Fourier transforms of these correlation functions it is possible toobtain the lineshape functions (Kubo, 1969). Following the sameapproach, we consider a frequency trajectory of the ‘‘excited state’’ coher-ence Oab(t):

Oab tð Þ ¼ ho0ai þ do0a tð Þ½ � � ho0bi þ do0b tð Þ½ �¼ hOabi þ do0a tð Þ � do0b tð Þ (13)

and the corresponding time correlation function:

Cab tð Þ ¼ h do0a tð Þ � do0b tð Þ½ � do0a 0ð Þ � do0b 0ð Þ½ �i; (14)

which can be expanded as:

Cab tð Þ ¼ hdo0a tð Þdo0a 0ð Þi þ hdo0b tð Þdo0b tð Þdo0b 0ð Þi� hdo0a tð Þdo0b 0ð Þi � hdo0b tð Þdo0a 0ð Þi: (15)



The first two terms are simply the same correlation functions that deter-mine the lineshape of fundamental coherences. The second two terms,however, are frequency–frequency cross-correlation functions. The line-shape of the excited state coherence in principle contains additionalinformation since it reports the degree of mutual correlation betweentwo different transitions. In the limit of fully un-cross-correlated fluctua-tions, the coherence line width should equal that of the fundamentalcoherences. Conversely, in the fully cross-correlated limit, the excitedstate coherence may be much narrower than the line widths of the funda-mentals (Engel et al., 2007). In the case of the system described here,DMDC, the lineshape of the excited state coherence is very similar tothat of the underlying transitions, suggesting that the two modesinvolved fluctuate essentially independently, despite their sharing ofidentical local CO units.

Figure 26 shows the isolated 62-cm�1 coherence with a simulation ofthe feature using the result of Equation (10) multiplied by a biexponentialoverall damping that describes the orientational relaxation of the cross-peak amplitude. The parameters of the simulated cross-peak amplitudeswere a 0.33 ps�1 coherence dephasing rate and a fast 5.2 ps�1 and a slow0.019 ps�1 damping. Since the scanned waiting time was not sufficient to

2 4 6t (ps)

80

0.1

Cro

ss p

eak

mag

nitu

de (

a.u.

)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

10

FIGURE 26 Trace (blue) of the cross-peak at o1 ¼ 1985 cm�1, o3 ¼ 2045 cm�1 showing

clearly the coherence signal. The data are reproduced well using a single coherence

dephasing rate of 0.33 ps�1, and a biexponential decay of the overall cross-peak

amplitude with rates of 0.019 and 5.2 ps�1



include the overall vibrational relaxation to the ground state, which isknown to be on the order of 100 ps, the biexponential relaxation is dueessentially entirely to orientational relaxation. The fast 5.2 ps�1 rate is dueto intramolecular vibrational redistribution, and leads to an orientationalrelaxation because the different transitions are oriented in different direc-tions, thus appearing as if the molecule were reorienting. The slower 0.019ps�1 rate arises from the diffusive reorientation of the whole molecule inthe liquid. The observation of vibrational wavepackets through thesecoherent oscillations serves to highlight the inherently three-dimensionalinformation contained in the full, third-order response function. Morerecently, we have used the observation of coherent cross-peakmodulationsto isolate the signature of chemical exchange in a flexible molecule with ahighly congested spectrum (Anna et al., 2009).

4. FUTURE DIRECTIONS

A growing number of research groups are employing 2DES for the studyof coupled electronic transitions in condensedphase systems ranging fromlight-harvesting systems to quantum-well structures. As the technicalbarriers to implementing 2DES continue to fall and further applicationsprove the utility of 2DES, this number will surely increase. In particular,the relative ease of transforming a pump-probe experiment into a 2DESexperiment may speed the process (Grumstrup et al., 2007; Myers et al.,2008; Shim et al., 2006; Shim & Zanni, 2009; Tekavec et al., 2009).

While numerous polarization 2DIR experiments have been performed,the use of polarization control has been limited to a handful of 2DES experi-ments (Read et al., 2008; Zhang et al., 2007). This degree of freedompromisesenhanced structural information and the ability to suppress unwantedsignal contributions. Alternative pulse sequences that probemultiple quan-tum coherences promise new ways to separate Liouville-space pathwaysand isolate spectroscopic signals sensitive to electronic correlations inmany-electron systems (Li et al., 2008; Yang &Mukamel, 2008).

Although the visible frequency regime currently represents the high-frequency limit at which 2D spectroscopy has been experimentally rea-lized, it is unlikely to remain that way for long. Theoretical work isalready showing the promise of extending 2D spectroscopy into theultraviolet regime (Abramavicius et al., 2006), which accesses the elec-tronic resonances of the basic constituents of proteins and DNA,promising new structural probes of conformational dynamics and energyand charge transfer in biological systems. At ultraviolet frequencies, thetechnical challenges of 2D spectroscopy can be readily met by pulse-shaping methods, so it is likely that 2DUV spectroscopy is on the horizon.Looking even further ahead, multidimensional X-ray spectroscopies for



exploring coupling between core-electron transitions have also been pro-posed and may become feasible with the development of high-intensitycoherent X-ray sources (Schweigert & Mukamel, 2007).

In the decade since two-dimensional spectroscopy was first demon-strated in the visible and infrared spectral regions, there has been sub-stantial progress in its application to condensed phase dynamics. Theoverwhelming majority of experiments have focused on systems thatare at equilibrium. That is, although the optical spectroscopy is by defini-tion nonequilibrium, it is viewed as a small perturbation permitting anearly noninvasive probe of equilibrium fluctuations. An alternative ave-nue has also been pursued using photoswitches (Hamm et al., 2008),temperature jumps (Chung et al., 2007), and photochemical (Baiz et al.,2008) and photophysical (Bredenbeck et al., 2004) triggers to use 2DIRspectroscopy as a probe of an inherently nonequilibrium process. Theseexperiments are essentially higher-order optical processes where an addi-tional correlation enables comparison of a differential 2D spectrum in thepresence and absence of some auxiliary optical modulation. It is a long-standing challenge in molecular spectroscopy and electronic structure toobtain detailed information about the vibrational potential surface ofexcited electronic states. Using 2DIR spectroscopy as a probe of dynamic,nonequilibrium excited electronic states will enable application of thepowerful spectral decongestion already demonstrated at equilibrium.These new approaches provide access to a vast range of timescales,offering site-specific, ultrafast dynamical snapshots of, for example,large-scale structural changes in a protein as it switches from an activeto a passive state. With the wider adoption of pulse-shaping techniques,wavelength regions can be combined yielding essentially arbitrary com-binations of UV, visible and IR frequency axes in novel multidimensionalspectroscopy methods. Key problems in light harvesting, organic light-emitting diodes, photocatalysis, as well as biological photochemistry areoften found to possess and exploit transitions in several spectroscopicwindows, and a combined approach will be needed to understand anddesign new materials as we search for replacements to finite energyresources.

ACKNOWLEDGMENTS

Wewish to thank our research groups for their creativity, dedication, andinspiration. JPO acknowledges the support from the Department ofEnergy, the National Science Foundation, the Alfred P. Sloan Foundation,and the American Chemical Society Petroleum Research Fund. KJKacknowledges the support from the National Science Foundation andthe American Chemical Society Petroleum Research Fund.



APPENDIX: DERIVATION OF THE T2-DEPENDENTCOHERENCE

This density matrix evolves freely according to the matter Hamiltonian,H0, until t3, when the third pulse arrives. Immediately before the thirdpulse arrives, the density matrix is:

rðcoherentÞ t3�ð Þ ¼ G0 t2ð Þ0 0 00 0 rab0 rba 0

0@

1A

¼ y t2 � t1ð Þ exp � i

�hH0 t2 � t1ð Þ

� � 0 0 00 0 rab0 rba 0

0@

1A; (A.1)

where G0 is the field-free Green function, y is the Heaviside function, andthe subscript ‘‘3–’’ on the time indicates this is the time just before t3.Written in terms of frequency differences ojk ¼ Ej � Ek

� �=�h, the propaga-

tor can be inserted into the matrix giving:

rðcoherentÞ t3�ð Þ ¼0 0 00 0 e�ioabðt2�t1Þrab0 eioabðt2�t1Þrba 0

0@

1A: (A.2)

The effect of the next pulse is another application of the dipole opera-tor, and the signal is proportional to the expectation value of the dipoleoperator. Since we only consider here the rephasing pathway, the thirddensity operator acts from the right according to the double-sidedFeynman diagram in Figure 7.

hm t3ð Þi ¼ Trfm½Gðt3ÞrðcoherentÞðt3�Þm�g

¼ Tr m0 0 0

eio0aðt3�t2Þe�ioabðt2�t1Þmb0rab 0 0eio0bðt3�t2Þeioabðt2�t1Þma0rba 0 0

0@

1A

24

35; (A.3)

where m is given as

m ¼0 m0a m0bma0 0 0mb0 0 0

0@

1A (A.4)



yields

hm t3ð Þi ¼ Treio0bðt3�t2Þeioabðt2�t1Þma0m0brba þ eio0aðt3�t2Þe�ioabðt2�t1Þm0amb0rba 0 0

0 0 00 0 0

0@

1A: (A.5)

Fourier transform of the signal with respect to t3 yields peaks at o0a or o0b

that are modulated by the coherence frequency oab. The above treatmentclearly neglects all relaxation, and is merely intended to explain thelocation of themodulated cross-peaks for a rephasing spectrum. Choosingone of these peaks, that centered at o0a, the t2-dependence is:

S coherenceð Þ o0a; t2ð Þ / e�ioab t2�t1ð Þm0amb0rab: (A.6)

Following the same reasoning as above, the contribution to the signal atthe same cross-peak due to the path that returns to the ground stateduring t2 is:

SðgroundÞðo0a; t2Þ / ma0m0ar00; (A.7)

where r00 is the ground state density matrix element during t2. The totalsignal at the cross-peak is given by the sum of the two paths:

S totalð Þ ¼ S coherenceð Þ þ S groundð Þ ¼ e�ioab t2�t1ð Þm0amb0rab þ ma0m0ar00: (A.8)

Since the signal is of the form f fð Þ ¼ A ei’ þ B, not only do the real andimaginary parts of the signal oscillate as a function off, so does the absolutevalue. Depending on the relative amplitudes of the dipole moment matrixelements and the density matrix elements, the absolute value modulationswill have varying degrees of visibility as shown in Figure A.1.

To determine the correct relative amplitudes A and B, we must exam-ine in detail the terms rab and r00 by explicitly accounting for the first twopulse interactions. Again we consider the t2-coherence and t2-groundstate paths separately. Both pathways, since we restrict ourselves torephasing diagrams, begin with the dipole operator acting from theright on the initial density:

r t1þð Þ ¼~r00 0 0

0 0 0

0 0 0

0B@

1CA

0 m0a m0bma0 0 0

mb0 0 0

0B@

1CA

¼0 m0a~r00 m0b~r000 0 0

0 0 0

0B@

1CA;

(A.9)

2.0

1.5

1.0

Cro

ss p

eak

mag

nitu

de (

a.u.

)

0.5

0.00 1 2 3

f4 5 6

FIGURE A.1 Absolute value of the cross-peak signal for three different values (0.1, 0.5,

1.0) of the relative amplitude of the ground state and coherent pathways. For the 1:1,

the signal shows a distinct cusp which is observed experimentally. As the coherence

dephases, the oscillations become more symmetric



where t1þ indicates the density immediately after application of m; thisdensity will evolve according to the field-free Hamiltonian between t1 andt2 as

r t1ð Þ ¼ G0 t1ð Þ0 m0a~r00 m0b~r000 0 0

0 0 0

0B@

1CA

¼0 e�io0at1m0a~r00 e�io0bt1m0b~r000 0 0

0 0 0

0B@

1CA:

(A.10)

For the t2-coherence pathway, the second pulse acts from the left:

r t2�ð Þ ¼ m0 e�io0at1m0a~r00 e�io0bt1m0b~r000 0 0

0 0 0

0B@

1CA

¼0 0 0

0 e�io0at1m0ama0~r00 e�io0bt1m0bma0~r000 e�io0at1m0amb0~r00 e�io0bt1m0bmb0~r00

0B@

1CA:

(A.11)

Thus, we see that the terms we originally identified as rab and rba:

rab ¼ e�io0bt1m0bma0~r00; rba ¼ e�io0at1m0amb0~r00: (A.12)



Inserting these into the expression found for the signal

SðcoherenceÞðo0a; t2Þ / e�io0bt1e�ioabðt2�t1Þjm0aj2jm0bj2~r00 (A.13)

and we find that, indeed, the t1 dependence shows that this signal ispeaked at o1 ¼ o0b, corresponding to a cross-peak at (o0b, o0a).

Next, we consider the pathway where the system returns to the groundstate. Following the common initial dipole application from the right, weoperate with the second field from the right as well:

r t2�ð Þ ¼0 e�io0at1m0a~r00 e�io0bt1m0b~r000 0 00 0 0

0@

1A

m ¼e�io0at1m0a~r00 þ e�io0bt1m0b~r00 0 0

0 0 00 0 0

0@

1A;

(A.14)

and thus we identify the density matrix element, r00:

r00 ¼ e�io0at1 m0ama0~r00 þ e�io

0bt1 m0bmb0~r00 (A.15)

and insert it into the expression for the signal at o0a due to the groundstate propagation:

S groundð Þ o0a; t2ð Þ / ma0m0ar00¼ ma0m0a e�io0at1m0ama0~r00 þ e�io0bt1m0bmb0~r00

� �: (A.16)

Focusing attention on the specific cross-peak (o0b, o0a), we consider theterm centered at o0b during t1:

S groundð Þ o0a; t2;o1 ¼ o0bð Þ ¼ ma0m0am0bmb0~r00 ¼ jm0aj2jm0bj2~r00: (A.17)

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