IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ... Schlau-Cohen.pdfDigital Object Identiﬁer 10.1109/JSTQE.2011.2112640 spectroscopy has proved to be a valuable tool to examine the dynamics

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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS 1

Ultrafast Multidimensional Spectroscopy: Principlesand Applications to Photosynthetic Systems

Gabriela S. Schlau-Cohen, Jahan M. Dawlaty, and Graham R. Fleming

(Invited Paper)

Abstract—We present the utility of 2-D electronic spectroscopyfor the investigation of energy transfer dynamics in photosyn-thetic light-harvesting systems. Elucidating ultrafast energy trans-fer within photosynthetic systems is difficult due to the largenumber of molecules and complex environments involved in theprocess. In many spectroscopic methods, these systems appear asoverlapping peaks with broad linewidths, obscuring the details ofthe dynamics. 2-D spectroscopy is a nonlinear, ultrafast methodthat yields a correlation map between excitation and emissionenergies, and can track incoherent and coherent energy trans-fer processes with femtosecond resolution. A 2-D spectrum canprovide important insight into the structure and the mechanismsbehind the excited state dynamics. We review the principles be-hind 2-D spectroscopy and describe the content of a 2-D elec-tronic spectrum. Several recent applications of this technique tothe major light-harvesting complex of Photosystem II are pre-sented, including monitoring the time scales of energy transfer pro-cesses, investigation of the excited state energies, and determina-tion of the relative orientations of the excited state transition dipolemoments.

Index Terms—Four-wave mixing, nonlinear optics, ultrafastoptics.

I. INTRODUCTION

B IOLOGICAL systems consist of complex machinery ofmolecular components that combine to drive nonequi-

librium processes through time and space [1]–[3]. The precisemolecular principles and detailed molecular mechanisms ofmost biological processes remain elusive. The complexity ofmolecular constituents within biological systems continuesto challenge experimental and theoretical methods. Ultrafast

Manuscript received November 3, 2010; revised January 4, 2011; acceptedJanuary 21, 2011. This work was supported by the U.S. Department of Energy,by the Office of Basic Energy Sciences, the Office of Science under ContractDE-AC02-05CH11231, and the Division of Chemical Sciences, Geosciences,and Biosciences under Grant DE-AC03-76SF000098 (at Lawrence BerkeleyNational Laboratory and University of California Berkeley), and by the DefenseAdvanced Research Projects Agency under Grant N66001-09-1-2026. The workof G. S. Schlau-Cohen was supported by the American Association of UniversityWomen American Dissertation Fellowship. The work of J. M. Dawlaty wassupported by the QB3 Distinguished Postdoctoral fellowship.

G. S. Schlau-Cohen is with the Lawrence Berkeley National Lab, Berkeley,CA 94720 USA, and also with the Department of Chemistry, University ofCalifornia, Berkeley, CA 94720 USA.

J. M. Dawlaty is with the Department of Chemistry, University of California,Berkeley, CA 94720 USA, and also with the Quantitative Biosciences Institute,Berkeley, CA 94720 USA.

G. R. Fleming is with the Department of Chemistry, University of California,Berkeley, CA 94720 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTQE.2011.2112640

spectroscopy has proved to be a valuable tool to examine thedynamics underlying a variety of biological processes [3], [4].In recent years, multidimensional spectroscopy has emergedrapidly as a particularly powerful ultrafast technique to studythe ultrafast dynamics of complex systems [5]. The techniqueprovides a unique window into ultrafast nonequilibriumbehaviors of these systems.

An understanding of the photosynthetic apparatus is partic-ularly critical in the context of a growing need for renewableenergy sources. Photosynthesis powers essentially all life onearth [6]. We review recent advances in understanding the dy-namics of the initial steps of photosynthesis (absorption andtransport of energy) within the most abundant light harvester,the major light-harvesting complex of photosystem II (LHCII).Different implementations of multidimensional spectroscopyprovide relative advantages in accessing various aspects of theultrafast dynamics of photosynthetic light harvesting. We de-scribe the principles behind and implementations of 2-D spec-troscopy, including different pulse orderings and polarizationsfor multidimensional spectroscopy. This tutorial is intended toserve as an introduction for readers unfamiliar with the fields ofmulti-dimensional spectroscopy or photosynthetic complexes.For more detail, we refer readers to the following recent re-views [7]–[15].

II. PHOTOSYNTHETIC COMPLEXES

Photosynthetic light harvesting occurs within arrays ofpigment–protein complexes (PPCs), which complete absorp-tion and subsequent transport of photoenergy to a designatedlocation (the reaction center) for a charge separation step todrive chemical reactions [1]. This contrasts with a photovoltaiclight-harvesting mechanism, where the initial absorption eventis a charge separation, or the creation of an electron–hole pair,throughout the material. In the case of green plants, photosyn-thetic light harvesting occurs in subcellular structures, or or-ganelles, called chloroplasts. These are disclike objects, about10 μm in diameter and about 2 μm thick. Within the chloroplast,there is a membrane constructed from a lipid bilayer, called thethylakoid membrane, arranged into stacks about 500 nm in di-ameter [16]. The thylakoid membrane houses dense networks ofPPCs, generally grouped into photosystems of about 20 PPCs.LHCII, the PPC we will be focusing on here, is about 2 nmacross, and its structure as obtained from X-ray crystallographyis shown in Fig. 1(a) [17]. In photosynthesis, the absorptionevent generates an excited state on the chromophores withinthe antenna array. The excitation then migrates through the

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2 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS

Fig. 1. (a) Structural model from X-ray crystallography of trimeric LHCII.Each monomer contains 14 chlorophyll (eight Chl-a, green, and six Chl-b, blue)and four carotenoids held within a protein matrix. (b) Linear absorption spectrumof the Qy (S0 –S1 ) transition of LHCII at 77 K. Reprinted with permission fromthe Proc. Natl. Acad. Sci. USA Figure first appeared in [55].

PPCs to reach the reaction center in a process that generallylasts approximately 200 ps [18], [19]. Each energy transfer steptakes between hundred femtoseconds to many picoseconds [20].Because solar intensity is only (on average) 1 kW/m2 [6], inorder to be able to utilize all incident photons, photosyntheticorganisms have a high density of pigments. Inside the chloro-plast, 80% of the area of the thylakoid membrane is occupied bythe PPCs [16]. Inside the protein matrix of a given PPC, thereis approximately 1 molar concentration of chlorophyll, whichis much more concentrated than the point at which quenchingeffects begin for chlorophyll in solution [17], [21]. The densityof chromophores within LHCII is illustrated in the structuralmodel in Fig. 1(a) [17]. As seen in Fig. 1(a), chlorophyll, theprimary light-harvesting molecule, and carotenoids, a secondarylight harvester, are held in a close-packed arrangement by aprotein scaffold. The density produces strong Coulombic in-teractions (10–500 cm−1) between chromophores and betweenchromophores and the protein [22]. The electrodynamic interac-tions are highly sensitive to position and relative orientation ofthe molecular components; therefore, the nanostructure of thechromophores within the protein matrix becomes critical for ef-fective energy transport [23]. The dense packing and resultantstrong Coulomb coupling between pigments (stronger than thecoupling of the pigments to the protein environment) drive ul-trafast energy transfer, produce delocalized excited states, andgive rise to quantum coherence, or superpositions between theexcited state wavefunctions.

The site basis Hamiltonian for the system is constructed fromthe uncoupled transition energies of the individual chlorophyll(site energies or diagonal elements), and the couplings betweenchromophores (off-diagonal elements). Diagonalization of thisHamiltonian results in delocalized excited states or excitons,which are linear combinations of the excited states of the indi-vidual chromophores [20]. Excitons are the energy eigenstatesof the system, and therefore, are the spectroscopic observables.A single exciton may contain contributions from multiple chro-mophores. Thus, the energies and transition dipole moments ofthe excited states result from the interactions between multiplemolecules within the structure.

This relation between the states that interact with light and themolecular site states, along with the effect of the surrounding

protein matrix, obfuscates the nature of the energy landscape ofa PPC within which energy transfer occurs. Pigment–pigmentcoupling terms, strength of the electron–phonon interaction, sizeof slow fluctuations of the protein structure, and broadening dueto fast vibrational motion all contribute to the energy trans-fer dynamics and spectroscopic measurements [20]. This largenumber of structural variables together determine the excitedstate energies and mechanisms of energy transport, and all havesimilar magnitudes making it difficult to use approximate mod-els [13], [20]. The collective effect on the energy landscape dueto these variables as measured by linear absorption is illustratedfor a model dimer of chlorophyll in Fig. 2. In Fig. 2(a), a sin-gle transition energy in the absence of any broadening variables(shown on the right) is shown for identical chlorophyll (shownon the left). In this case, the excitation Hamiltonian is simply

Hel =N∑

j=1

|j〉 ε0 〈j| (1)

where ε0 is the transition energy of a pigment and is the same forall pigments j. In PPCs, chlorophylls are held within a proteinmatrix, and the protein pockets surrounding each chlorophyllmolecule vary (illustrated in Fig. 2(b), left). These differenceschange the protein environment and, therefore, change the tran-sition energies of the chlorophyll. As shown in Fig. 2(b), right,the single transition has changed to two. The Hamiltonian canthen be written as

Hel =N∑

j=1

|j〉ε0j 〈j| (2)

where ε0j is the transition energy of the jth pigment. As discussed

above, the dense packing of chlorophyll in PPCs leads to strongcouplings (shown in Fig. 2(c), left), thus changing the transitionenergies and dipole strengths (reflected in Fig. 2(c), right). Withthe addition of coupling terms, the Hamiltonian becomes

Hel =N∑

j=1

|j〉ε0j 〈j| +

N∑

i<j

Jij (|i〉〈j|+ |j〉〈i|) (3)

with coupling energy Jij between sites i and j. Wavefunctionsof the diagonal form of this Hamiltonian or excitons are linearcombinations of single chromophore excitations

|ψk 〉 =N∑

j=1

ckj |j〉 (4)

where the coefficient ckj is proportional to the contribution ofchromophore j to exciton k. Diagonalization of Hel also givesrise to multiply excited states, where more than one excitationresides in the complex at the same time. In this work, we willrestrict most of our discussion to the one exciton states becausethese are the only states accessed by the low density of sunlight.The two-exciton states can, however, contribute to a 2-D spec-trum and can provide important insight into coupling betweenpigments [24], [25], as will be discussed later. Because mostspectroscopic experiments are ensemble measurements, mean-ing the volume of the sample interrogated by the laser contains


SCHLAU-COHEN et al.: ULTRAFAST MULTIDIMENSIONAL SPECTROSCOPY 3

Fig. 2. (a) Two noninteracting chlorophylls without a surrounding environ-ment (left) exhibit a single, narrow resonance (right). (b) Two noninteractingchlorophylls in a static surrounding protein environment (left) give rise to twonarrow resonances (right) due to differences in the protein surrounding thetwo chlorophylls. (c) Two interacting chlorophylls in a static protein environ-ment (left) give rise to two narrow resonances at different energies (right) dueto increased splitting from their coupling. (d) Ensemble measurements of aslowly fluctuating sample (left) give rise to an inhomogeneous distribution ofresonances (right). (e) Vibrational motion (right) gives rise to homogeneousbroadening (right).

a large number of complexes, the variations between complexesdue to slow fluctuations in protein configuration lead to inho-mogeneous broadening in the transition energies. Configurationchanges mean differences in orientation and distances betweenthe chromophore transition dipoles relative to the protein back-bone and to the polar groups bound to the protein, changes in

bowing of the aromatic ring of the chlorophyll, etc. These fluc-tuations change the chromophores’ transition energies (and mayaffect the coupling). The ensemble is shown in Fig. 2(d), left, andbroadening from variations in the ensemble in Fig. 2(d), right.Thus, to represent a given measurement taken on an ensemble ofPPCs, the calculated dynamics must be averaged over a distribu-tion of transition energies ε0

j . Finally, high frequency vibrationsof the protein environment lead to homogeneous broadening,producing the spectrum shown in Fig. 2(e), right. In the pres-ence of the protein environment, the Hamiltonian becomes

Htot = Hel + Hph + Hel−ph . (5)

The dynamics of the phonon modes of the protein bathare contained within Hph . For the electron–phonon couplingHel−ph , a displaced harmonic oscillator model is used [10], [26].These couplings produce site energy fluctuations described as

Hel−ph =N∑

j=1

(−

∑

ξ

h̄ωξdjξ qξ

)|j〉〈j| . (6)

The electron–phonon coupling term for each pigment j is asum over all phonon modes, or all vibrations within the proteinbath, where the ξth mode is described by the dimensionless vi-brational coordinate qξ and frequency ωξ . The strength of thecoupling is scaled according to a proportionality constant dξ ,corresponding to the displacement of the excited state potentialenergy surface along the generalized coordinate qξ . [10], [13]This large number of variables contributing to the system Hamil-tonian (and consequently to measurements) makes it difficult toidentify the contribution of each component to the dynamics.For example, the linear absorption shown for the model dimerlooks almost indistinguishable from a single transition with ahigh energy tail.

LHCII, the complex discussed in this study, is the powersource for green plants. It binds over 50% of the world’s chloro-phyll and is the most abundant light-harvesting complex [27].As shown in Fig. 1(a), LHCII has a trimeric structure. Eachmonomer contains 14 chlorophylls and 4 carotenoids held withina protein matrix of three fully and one partially transmembranealpha-helices [17]. The 14 chlorophylls are found in two spectraland structural variants, 8 Chl-a and 6 Chl-b. The linear absorp-tion spectrum of the Qy (S0–S1) transition of the chlorophyllis shown in Fig. 1(b). The spectrum is characterized by twobroad peaks, with Chl-a centered at ∼14 900 cm−1 and Chl-b at∼15 400 cm−1 .

The dynamics of light harvesting in LHCII are difficult tostudy by many conventional methods due to broad overlappingtransitions and a variety of time scales of energy transfer [28].Development of advanced spectroscopic methods is necessaryto learn about the initial steps of photosynthesis. In the followingsection, we present 2-D electronic spectroscopy as an emergingtechnique for this purpose.

III. FUNDAMENTALS OF 2-D ELECTRONIC SPECTROSCOPY

A variety of ultrafast spectroscopic methods have been usedto investigate the range of ultrafast dynamics of the initialsteps of photosynthesis (absorption and transfer of energy in



photosynthetic antenna complexes). Examples of suchmethods are pump–probe spectroscopy, fluorescence line-narrowing, time-resolved fluorescence, hole-burning, polariza-tion anisotropy decay, photon-echo peak shift spectroscopy, etc.Many of these techniques are third-order nonlinear optical spec-troscopies, where the response of the sample to the third powerof light electric field is measured. A generalized classificationof a variety of ultrafast nonlinear spectroscopy techniques canbe found in many comprehensive references [26], [29], [30].A large range of dynamical information is embedded in thethird-order nonlinear response of the photosynthetic complexes.However, depending on experimental configuration, the abovetechniques have access to limited views of the full range ofthird-order dynamics. We will clarify this shortly, by discussingthe conventional pump–probe spectroscopy technique as anexample.

In this section, we argue that 2-D electronic spectroscopyis particularly distinguished from the aforementioned methods,as it aims to measure the full complex degenerate four-wave-mixing third-order nonlinear response of photosynthetic com-plexes within their relevant absorption bandwidths. It can beviewed as a generalized approach that not only subsumes manyconventional third-order spectroscopy techniques, but also re-veals an abundance of information that is otherwise inaccessibleby the other methods. To make this distinction clear, we writethe nonlinear polarization induced in a sample due to interactionwith three laser pulses [26]:

P (3)(r, t) =∫

dt3

∫dt2

∫dt1S

(3)(t3 , t2 , t1)

× E3(r, t− t3)E2(r, t− t3 − t2)E1(r, t− t3 − t2 − t1)

(7)

where Ej (r, t) = Aj (t){e−i(ωt+kj ·r) + ei(ωt−kj ·r)} are thepulse electric fields with envelopes Aj (t), central frequencyof ω and wave vectors kj . The third-order nonlinear responseof the material S(3)(t3 , t2 , t1) relates the driving fields at timest1 , t2 , and t3 to the induced nonlinear polarization at time t. Theresponse function contains information about the relevant quan-tum states of the sample, all possible optical transitions betweenthem at times t1 , t2 , and t3 and their free evolution when theyare not interacting with the fields.

To distinguish 2-D spectroscopy from other conventionalmethods, we discuss pump–probe spectroscopy in the context ofthe aforementioned equation. In pump–probe spectroscopy, thefirst two interactions with the electric field are derived from thesame optical pulse (pump); hence, any time, spatial directionand polarization resolution between the interactions are lost.Furthermore, the last interaction (probe) and the nonlinear sig-nal emitted by the polarization P (3) share the same direction.The interference of the probe with the signal results in changesof the probe intensity and the phase of the signal is not mea-sured. In fact, the first two interactions also occur at the intensitylevel (EpumpE∗

pump ), and the experiment is interpreted as theeffect of the intensity of the first pulse on the transmission of theprobe. Common oscillatory behavior seen in pump–probe tran-sients are usually due to coupling of the electronic transitions to

Fig. 3. (Top panel) Excitation pulses (E1 –E3 ) and the reference pulse forphase retrieval Eref used in 2-D spectroscopy, shown both in time and spatialdirections with respect to the sample. (Middle panel) Representative Liouvillespace response pathways to the complex components of the driving electricfields. Solid (dashed) lines show evolution of the bra (ket) sides of the densitymatrix. (Bottom panel) Sum overall coherences oscillating at optical frequen-cies. The oscillating polarization of the last coherence results in signal emissionin the ks direction.

other modes, such as impulsively driven vibrations, and do notreflect direct measurement of the phase of the electronic tran-sitions. Another drawback of the pump–probe technique is thatan increase in spectral selectivity by narrowing the bandwidthof the pulses is necessarily accompanied by a loss of resolutionin time. Even when the probe is spectrally resolved, intensitychanges of a given frequency component of the probe cannot becausally linked to the action of particular frequency componentsof the pump.

In the following, we will discuss how 2-D spectroscopy candistinguish between the three interactions in time, space, andpolarization. Both the phase and amplitude of the nonlinear po-larization is measured and a frequency–frequency correlationis built between the first and the last interaction without lossof temporal resolution. The excitation pulses and the compo-nents of S(3) that are relevant to 2-D electronic spectroscopyare shown in Fig. 3. A model material system with two excitedstates similar to the exciton structure of the chlorophyll dimer(see Fig. 2) is considered. The three driving electric fields E1–E3are time delayed by times τ and T. The fourth pulse Eref is usedas a phase reference for retrieving the phase of the nonlinear sig-nal by spectral interferometry, as is routinely performed in manyapplications within ultrafast optics. Phase retrieval is discussedin detail in the context of 2-D spectroscopy in references [5] and[31]–[33]. In brief, the reference pulse has a stable phase withrespect to the third pulse and is heavily attenuated to avoid itsnonlinear interaction with the sample. It is guided to a spectrom-eter along with the copropagating signal. The interference of the



signal and the pulse appears as spectral fringes with an oscilla-tory component cos(Δφ(ω)), where the phase difference Δφ(ω)between the reference and the signal is retrieved by Fouriertransformation.

While the temporal phase ±iωt of the pulses induce res-onant transitions between quantum states in the sample, thespatial phase ±ik · r of the beams determine the phase dif-ference between these transitions across the focal volume inthe sample, resulting in spatial phase gratings of excitations.Due to the interferences of such gratings induced by the threepulses, the nonlinear polarization P(3) contains many spatialfrequencies (all permutations of ±k1±k2±k3) that radiate intotheir corresponding spatial directions. By choosing to measurethe signal in the ks = −k1 + k2 + k3 direction, we only se-lect for a particular set of these pathways. Such spatial phasematching, also known as momentum conservation, has two ad-vantages. First, the signal appears in a background-free direc-tion in space, unadulterated by the strong excitation beams.The second advantage is that the signal only contains contri-butions from molecular transitions induced by the correspond-ing ordered combination of frequencies +ω1−ω2−ω3 (energyconservation).

2-D spectroscopy can also be performed with either all or twoof the excitation pulses sharing the same direction k, known asfully and partially collinear geometries [34]–[36]. In those cases,to isolate the signal, it becomes necessary to have control overthe temporal phases of the pulses via a pulse shaper. In a processknown as phase cycling, the signal is isolated due to its uniquephase dependence on all the three excitation pulses [34], [37].As seen in (7), the nonlinear signal is related to a multipli-cation of the three excitation electric fields. Multiplying anyodd (even) number of pulses by a factor eiπ , using a pulseshaper, will flip (not flip) the phase of the signal. If the sumof all odd permutations of phase flips is subtracted from thesum of all even permutations, the resultant signal will be de-pendent on all the three excitation pulses and, hence, free fromthe overwhelming lower order background. Phase cycling canalso be used in the fully noncollinear geometry as an extrameasure of signal discrimination and removal of linearly scat-tered light [38], [39].

A. Liouville Space Pathways

Since the phase-matching condition requires us to differen-tiate between the interaction of the material with the two con-jugate components of the electric field (negative and positivefrequencies) separately, it becomes necessary to distinguish be-tween the two conjugate components of the electronic excita-tions as well. To that end, we can represent the state of thesystem using the density matrix formalism, which tracks allquantum states, their superpositions and their conjugates in asingle ensemble-averaged entity. Since this formalism is dis-cussed in detail in many texts [26], [40], we will give only abrief introduction in the context of 2-D spectroscopy. The den-sity matrix element ρab is given as

ρab =1N

∑

n

|a〉〈b|. (8)

When the nth member of an ensemble is in the superpositionstate |ψ〉n = ca |a〉 + cb |b〉, the matrix element of the projec-tion operator |a〉〈b| will be the complex quantity cacb∗, carry-ing information about both the magnitude and the phase of thesuperposition state. This complex quantity is further averagedover the ensemble of N members, and will be only nonzero ifthe quantum superpositions within the ensemble have a distinctphase relation with each other. The quantity ρab , known as theab coherence, is thus a measure of phase coherence of super-positions over the ensemble. The projector |a〉〈a| retrieves themagnitude of the contribution of state a or |ca |2 . Its ensemble av-erage ρaa is independent of the relative phases of the moleculesand is equal to the ensemble-averaged population of state a. Tocalculate the third-order nonlinear response of the system (7),the density matrix evolution equation (the Liouville equation)is evaluated perturbatively up to the third order in the electricfield of light [26].

Since the density matrix contains all coherences and theirconjugates, optical transitions between states induced by thenegative and positive frequencies of the incident fields can beeasily followed. A positive frequency causes excitation (de-excitation) when acting on the bra (ket) sides of a density matrixelement, respectively, provided the transition is resonant withthe incident frequency. Similarly, a negative frequency causesexcitation (de-excitation) when acting upon the ket (bra) side ofa density matrix element. A series of such transitions betweenthe elements of the density matrix induced by complex electricfields is known as a Liouville space pathway, and is often rep-resented in the form of double-sided Feynmann diagrams [26].Each Liouville space pathway also carries a positive (negative)sign, if it has an even (odd) number of interactions from the braside [26]. The laser pulses used in 2-D spectroscopy have a largebandwidth in frequency to allow simultaneous excitation of alloptical transitions in the sample and to ensure narrow temporalwidths. Due to this large frequency bandwidth, a multitude ofLiouville space pathways with different frequency combina-tions become possible. The final radiating polarization is theresult of summation over all of those complex Liouville spacepathways that satisfy the phase-matching condition. Isolationof overlapping pathways is a recurring problem in nonlinear op-tical spectroscopy. 2-D electronic spectroscopy and its variantsdiscussed here have proven successful in extricating groups ofpathways based on time ordering, spatial phase matching, andpolarization.

Three representative Liouville space pathways are graphicallydemonstrated in Fig. 3. Vertical solid (dashed) arrows representoptical transitions induced on the bra (ket) sides of the quan-tum states. The first path demonstrates a series of interactions+ωb−ωb−ωb with pulses 1, 2, and 3, respectively. The first pulseprepares the system in a superposition state |g〉〈b|, which freelyevolves for time τ . The second pulse converts that coherence intopopulation state |b〉〈b| that lasts for time T. The last pulse actson the bra side of this coherence and changes the population intoa coherence once again, which results in a radiating polariza-tion in the phase-matched direction. The second pathway arisesfrom the interaction of frequencies +ωa−ωb−ωa with pulses1, 2, and 3, respectively. Since the first and second interactions



Fig. 4. (Left) Electric field of the emitted signal plotted as a function of tand τ . As τ is scanned, the phase of the emitted signal (red line) oscillates,revealing the correlation of the signal to coherences present during τ . Thiscorrelation between the excitation and emission frequencies is better representedby Fourier transformation of the signal along the t- and τ -axes, resulting in a2-D spectrum (right). Representative Liouville space pathways contributing todiagonal and off-diagonal peaks are shown. Excited state coherences (circled),which result in slow phase oscillations during T, contribute to the diagonal peaksvia nonrephasing pathways and to off-diagonal peaks via rephasing pathways.

have different frequencies, the pathway evolves as a coherence|b〉〈a| (superposition between states a and b) during the waitingtime. The relevance of such pathways to understanding coherentdynamics of excitons in photosynthetic complexes will be dis-cussed shortly.

B. 2-D Electronic Spectrum

A 2-D spectrum, for a given waiting time T, is a correlationmap between the coherences during time τ and the radiatingoptical coherences during time t. For example, two representa-tive Liouville space pathways in Fig. 3 emit light at frequencyωb , although they arise from different excitation coherences atfrequencies ωa and ωb during time τ . Since emission from eachpathway simply add to emission from other pathways, it is notpossible to discern the relation of the emitted light to the excita-tion coherences from a single snapshot of the emitted field. Toexperimentally measure this correlation, it becomes necessaryto acquire the emitted signal for a range of delay times τ , asshown in Fig. 4(a). As τ is scanned, the phase of the emittedsignal (dashed line in Fig. 4) oscillates at the frequencies ofthe contributing coherences during τ . Such correlation can beconveniently depicted, if the signal is Fourier transformed alongboth the τ - and t-axes, resulting in a 2-D electronic spectrum(see Fig. 4). Liouville space pathways that contribute to a rep-resentative diagonal and off-diagonal peak are also shown inFig. 4. For clarity, the pathways that evolve on the ground stateduring the waiting time and the pathways corresponding to thetwo-exciton manifold are not shown.

It should be kept in mind that the pathways involving the two-exciton manifold have significant impact on the appearance of2-D spectra of photosynthetic complexes and contain importantinformation related to the spatial overlap of the excitons (andhence coupling between pigments). A representative pathwaythrough the two-exciton manifold is shown in Fig. 3, where thelast coherence is between the two-exciton state f and state a. Ifthe transitions a and b reside spatially far away from each other(e.g., on two uncoupled pigments), the state f can be accessedby two independent transition frequencies ωa and ωb (i.e., theenergy gap between f and a will be exactly equal to ωb ). In that

case, the coherence |f〉〈a| will radiate at the same frequency asthe other off-diagonal Liouville space pathways in the figure. Itshould be noted that this pathway has an overall negative sign(one interaction on the bra side), and its addition to the otheroff-diagonal pathway will result in complete cancellation of thecorresponding cross peak. However, if the pigments are coupledand the transitions ωa and ωb are delocalized excitons, the energyof the state f is modified due to the interaction of the two over-lapping excitons. Due to the frequency offset of the coherence|f〉〈a| from the coherence |b〉〈g|, only a partial cancellationwill occur. This cancellation of Liouville space pathways resultin relatively small cross peaks in the 2-D spectra of photosyn-thetic complexes near zero waiting time. As will be explainedshortly, at larger population times, cross peaks emerge belowthe diagonal of the 2-D spectrum due to population transfer.

Finally, it should be noted that a 2-D spectrum is a complexquantity. As mentioned earlier, the phase of the emitted sig-nal is retrieved with respect to a reference pulse. However, inmany experimental implementations, the phase relation of thereference pulse with respect to the first two excitation pulsesis unknown. A commonly used method to establish the phaserelation of the signal to the first two pulses is based on requiringthe projection of the real part of the 2-D spectrum on the emis-sion axis to match with an independently acquired spectrallyresolved pump–probe measurement. Further discussion on thismethod can be found in [5].

C. Rephasing and Nonrephasing Pathways

As seen in Fig. 4, in some pathways, the excitation coher-ence (during τ ) and emission coherence (signal) occur on theopposite sides of the density matrix. Such pathways, where theinitial and final coherences are evolving with opposite signs offrequency are known as rephasing pathways. The origin of thename rests in the fact that the ensemble-averaged coherenceof an inhomogeneous distribution of frequencies during time τwill decay due to the variation in their frequencies. However,during t, where all of those frequencies evolve with the oppo-site sign, such dephasing is undone and the decayed coherencereemerges. The reemergence of the decayed coherence is knownas photon echo and bears direct resemblance to the equivalentspin echo in magnetic resonance spectroscopy. In contrast, inthe nonrephasing pathways, the initial and final coherences oc-cur on the same side of the density matrix, and the rephasing ofthe excitation coherence does not occur. It can be verified fromFig. 4 that for a nonrephasing pathway, the order of interactionof the first two positive and negative frequencies must change.Experimentally this can be achieved by scanning τ to negativevalues, and hence, swapping the time ordering of pulses 1 and2, which does not affect the phase-matching direction for thesignal. By restricting the range of the delay time τ to positive(negative) values, a rephasing (nonrephasing) 2-D spectrum canbe constructed separately. When both positive and negative de-lay times are considered, the resulting 2-D spectrum is alsoknown as a total or relaxation spectrum.

Two experimental implementations of 2-D spectroscopy areshown in Fig. 5. In Fig. 5(a), a diffractive optic generates pairs



Fig. 5. Two implementations of 2-D spectroscopy. (a) Diffractive-optic-basedapparatus [73]. Two pulses (generated by a beam splitter and delayed by aretroreflector on a stage) are focused onto a diffractive optic, optimized for±1 orders, to generate two pulse pairs. The four beams are focused onto thesample, with beams 1 and 2 delayed with attosecond precision by glass wedges.The signal emerges phase-matched with beam 4 and is spectrally dispersedand detected in the frequency domain on a CCD camera. (b) Pulse shaperimplementation based on [39]. Four pulses are derived from an expanded laserbeam using a mask. They go through a focus and then to a 4f pulse shaper witha reflective 2-D SLM. The delayed and shaped pulses are retrieved from thefocus and used for the experiment.

of phase-stable pulses. Glass wedges and retroreflectors on me-chanical stages are used to introduce delays between them [31].The apparatus in Fig. 5(b) generates four beams using a spatialmask and delays are introduced by a spatial light modulator(SLM) in a 4f pulse shaper geometry [39].

D. Dynamics During Waiting Time

Acquisition of 2-D spectra as a function of waiting time Tprovides insight into the dynamics within the excited state ofthe photosynthetic complexes. Here, we will describe at leastthree types of dynamical processes during the waiting time thatare studied by 2-D spectroscopy in photosynthetic complexes.

1) Spectral Diffusion: As explained earlier, a 2-D spectrumcorrelates frequencies of coherences during times τ and t. Atransition frequency, for example ωa , can be inhomogeneouslybroadened over the ensemble due to variations of local environ-ments of the chromophores. As long as these local environmentsare unchanged between the excitation and emission times, everymember of the ensemble will emit at the same frequency thatit was excited with. Such a correlation between the excitationand emission frequencies, results in diagonally elongated ellip-

tical line shapes for the diagonal peaks. However, as time T isincreased, the frequencies of the transitions can change due tofluctuations of the local environments. As 2-D spectra are ob-tained for larger and larger population times, the ellipticity ofthe diagonal peak, which is the evidence of frequency–frequencycorrelation between excitation and emission coherences, is grad-ually washed away. Measure of the ellipticity of the diagonalpeak provides, providing a direct window into fluctuating envi-ronments of chromophores.

2) Population Transfer: Spatial transfer of excitation energyin photosynthetic light-harvesting complexes is usually accom-panied by stepping down an energy ladder, where energy istransferred from a higher energy exciton to a lower energy one.The signature of such transfer conveniently appears in 2-D elec-tronic spectroscopy as the emergence of cross peaks between thehigher energy excitons during τ and lower energy excitons dur-ing t. The emergence of such cross peaks due to excitation energytransfer has been studied in many photosynthetic complexes us-ing 2-D spectroscopy [28], [41]–[45]. In particular, nonrephas-ing 2-D spectra have more advantages for detecting populationtransfer, as there are no other direct nonrephasing pathwaysthrough the single-exciton manifold that generates cross peaks(see Fig. 4). It should be emphasized that the information onincoherent population transfer obtained from 2-D spectroscopyis more complete than that of a spectrally resolved pump–probeexperiment. The 2-D nature of the spectrum correlates the ini-tial and final energies (cross-peak coordinate) of the migratingexcitation. In contrast, a spectrally resolved pump–probe tracecan only determine the final energies, without causally linkingthe final and initial excitations to each other.

3) Coherence Quantum Beats: Figs. 3 and 4 show that whilesome Liouville space pathways evolve as population states (di-agonal elements of the density matrix, e.g., |a〉〈a|) during thepopulation time, some others evolve as coherences within theexcited states (off-diagonal elements of the density matrix, e.g.,|b〉〈a|). The free evolution of the off-diagonal element |b〉〈a|of the density matrix results in the acquisition of a phase atthe difference frequency between states a and b. This phase isdirectly imparted on the emission coherence from that pathway.Two such pathways that involve the free evolution of coherencesduring the population time are shown in Fig. 4(b), where the off-diagonal element of the excited state density matrix is markedby a circle. From the figure, it is easy to see that a rephasingpathway can only access an excited state coherence, if the initialand final coherences have different frequencies (a rectangular,rather than a square path within the density matrix), correspond-ing to an off-diagonal peak in the 2-D spectrum. Consequently,the off-diagonal peak at ωa and ωb will acquire an oscillatoryphase as a function of the waiting time T at the difference fre-quency of the two transitions ωa – ωb . An example of such phaseoscillations was recently shown in the Fenna–Mathew–Olson(FMO) complex [46]. On the other hand, many nonrephasingpathways can contribute to a single diagonal peak, with eachpathway experiencing a different excited state coherence dur-ing the population time. Consequently, a single diagonal peakwhen scanned as a function of population time, will acquire aphase proportional to all coherences within the excited state.



Interference between these pathways shows up as amplitudebeats of the diagonal peak as a function of population time.Such beats of the diagonal and off-diagonal peaks of a 2-Dspectrum at the difference frequency between excitonic tran-sitions are evident for coherent-free evolution of the excitonsduring the population time.They have been observed in mul-tiple experiments on the photosynthetic complexes FMO [45],LHCII [47], and phycobiliproteins [44].

E. Polarization-Dependent 2-D Spectroscopy

The ability to individually control the polarization direction ofeach one of the excitation beams [each one of the electric fields in(7)] adds additional power to 2-D spectroscopy. It is importantto distinguish polarization-dependent 2-D spectroscopy fromthose polarization-dependent third-order spectroscopies wheretwo interactions with electric fields are derived from the samebeam, and hence, forced to share the same polarization. In fullynoncollinear 2-D spectroscopy, it is possible to prepare polariza-tion sequences that cannot be achieved with partially collinearnonlinear optical methods. Orientational effects in spectroscopyhave long been recognized as an integral contributor to the de-tected signal [48], and a more complete description is availablein many books [49], [50].

Each light–matter interaction in a Liouville space pathwayis a dot product between the light electric field and the sampletransition dipole. Although the pigments in the protein are heldin fixed angles with respect to each other, the experiments areperformed on isotropic ensembles of proteins with no long-range order. For that reason, each Liouville space pathway isscaled by an orientational prefactor, which accounts for isotropicaveraging of all the four light–matter interactions in that pathway[49], [51], [52]. The orientational prefactor has the followingmathematical form:

〈iα jβ kγ lδ 〉 =130

[〈cos θαβ cos θγδ 〉 (4 cos θij cos θkl

− cos θik cos θjl − cos θil cos θjk )

+ 〈cos θαγ cos θβδ 〉 (4 cos θik cos θjl

− cos θij cos θkl − cos θil cos θjk )

+ 〈cos θαβ cos θγδ 〉 (4 cos θil cos θjk

− cos θij cos θkl − cos θik cos θjl)] (9)

where i, j, k, and l are the light polarizations in the lab frame andα, β, γ, and δ are the excited state transition dipole moments inthe molecular frame (the PPC frame) averaged over an isotropicdistribution. This expression arises from the fourth-rank tensorinvariants, or the pathways, which, when rotationally averaged,do not sum to zero. The relationships between the differentframes, which produce the aforementioned expression, can bewritten as [49], [51]:

〈iα jβ kγ lδ 〉 = (pαqβ rγ sδ )(aibj ckdl) 〈apbq crds〉 (10)

where the first term relates the dipole directions to an orthog-onal system (p, q,. . .) fixed within the molecular frame. Thesecond term relates the polarizations of the incident pulsesto an orthogonal system (a, b,. . .) fixed within the laboratory

frame, and the final term is the isotropic average over the lab-oratory and molecular fixed frames. The angle between thetransition dipole moments of the excited states participatingin a given Liouville space pathway directly affects the orienta-tional prefactor. This dependence can be exploited to decongestcrowded 2-D spectra by selecting out certain Liouville spacepathways [53]–[55]. A particularly useful polarization scheme isthe “cross-peak-specific” polarization sequence (π/3,−π/3,0,0),where the angles refer to the relative polarizations of the threeexcitation beams E1–E3 and the reference beam Eref , respec-tively. Since only parallel polarization can interfere and the sig-nal is retrieved through interferometry with Eref , selecting thepolarization of Eref also selects for the polarization directionof the signal. The utility of the cross-peak-specific polarizationsequence is that it suppresses pathways, that evolve as excitedstate populations without energy transfer during the waitingtime. As a specific example, consider the case where all transi-tion dipole directions in a Liouville space pathway are parallelto each other in the molecular frame. This is true, for exam-ple, when all interactions occur with the same dipole, such asthe diagonal pathway in Fig. 3. These pathways cannot be ac-cessed by the cross-peak-specific polarization sequence becausethe corresponding orientational prefactor is zero. This approachcan also suppress the signals from excited state absorption andincoherent energy transfer, while isolating the signal from path-ways that evolve as quantum coherences between nonparalleldipoles during the waiting time [53].

In addition to decongesting 2-D spectra, control over the po-larization of the incident laser pulses offers a means to determinethe geometry of the electronic structure (the relative directions ofthe exciton transition dipole moments) [55], [56]. Since excitonsare superpositions of many molecular transitions, their transi-tion dipole moment also becomes a vector sum of the dipolemoments of the contributing pigments. The excitonic transitiondipole moments, therefore, depend on site energies, couplings,and the relative orientations of the pigments and cannot be sim-ply determined from X-ray diffraction data. The geometry canbe inferred from the angles between the transition dipole mo-ments of the excitons. The angles between transition dipolemoments can be accessed by measuring the orientational pref-actor through a comparision of the amplitude of a peak undertwo polarization sequences. Measuring the orientational prefac-tor using two known polarization sequences of the incident laserpulses provides a measurement of the angle between transitiondipole moments of the excitons.

IV. SITE ENERGY DETERMINATION BY

THE EXCITED STATE GEOMETRY

Although the sample used for 2-D experiments usually con-tains an isotropic distribution of individual PPCs, within eachPPC the pigments are held in a fixed orientation by the pro-tein matrix. Therefore, the approach described in Section IIIcan be used to access the angles between the excitons of thepigments. The ultimate goal of these experiments on photo-synthetic complexes is to relate molecular structure to func-tion. However, relating structure to function, which is far from



straightforward. Structure determination experiments, such asX-ray diffraction, are done in conditions, which may differ sig-nificantly from physiological conditions. In addition, becauseof the large number of variables and strong interactions, it isdifficult to calculate a unique solution for how a given molecu-lar model gives rise to a set of excited states and their dynamics.Experiments using the incident pulse polarization offer an inci-sive way to probe the structure of the excited states, and thus,provide an important complement to structural studies. Dynam-ical models can be benchmarked against geometric informationprovided by measurements of angles between transition dipolemoments.

Determination of angles between excitonic transition dipolemoments using the approach discussed in the previous section isa sensitive probe to understand the link between the individualchlorophylls (sites) and the excitons, or between the molec-ular structure and the excited states. The excitonic geometrydepends on the site-basis transition dipole moments, the cou-pling between sites and the transition energies of the uncoupledchlorophylls (site energies). The first two variables can be cal-culated from the structure, but the site energies are not easilydetermined. Both the exciton energies and the directions of thetransition dipole moments depend on the same three variables.The exciton energies increase roughly linearly with increasingsite energies. The direction of transition dipole moments, how-ever, rotates based on changes in the mixing angle (from changesin the difference between site energies relative to the coupling).This rotation means that peak amplitude scales nonlinearly withsite energy so that in many cases, small changes of site energylead to drastic changes in angles between excitonic transitiondipole moments. This, in turn, produces large changes in ori-entational prefactor. Therefore, the direction of the excitonictransition dipole moment is a much more sensitive probe of thesite energies than is the exciton energy [55].

Fig. 6 displays absolute value, nonrephasing spectra ofLHCII under all-parallel and cross-peak-specific polarizationsequences. These spectra are characterized by three peaks alongthe diagonal: Chl-a peak at 14 775 cm−1 with a shoulder at14 900 cm−1 and, a Chl-b peak at 15 450 cm−1 . The peak in-dicated by an arrow in Fig. 6(a) and (b) shows energy trans-fer from the Chl-b states into the region between the Chl-band Chl-a bands. This peak is an example of the utility of thepolarization dependence, because it is both buried in the all-parallel spectra and can be used to determine site energies oftwo chlorophylls. The spectra in Fig. 6(a) and (b) show a cleardifference between relative intensities on the left (correspond-ing to all-parallel polarizations) and on the right (correspondingto cross-peak-specific) of the cross-peak highlighted by a whitearrow. Based on the scaling from (9), enhancement under thecross-peak-specific polarization indicates that the angle betweentransition dipole moments of the donor and acceptor states isclose to 90◦. Peaks from states with transition dipole momentsclose to parallel are suppressed, and peaks from states withtransition dipole moments close to orthogonal are enhanced.The enhancement of this peak, compared to the many otherpeaks in the spectra, means that the transition dipole momentsmust have an angle between them close to 90◦.

Fig. 6. (a) Nonrephasing, absolute value 2-D spectrum of LHCII at 77 K takenunder all-parallel polarization for T = 400 fs. The arrow highlights the Chl-b tointermediate region (between the Chl-a and Chl-b bands), which displays onlya shoulder of the midenergy Chl-a peak in this spectrum. (b) Nonrephasing, ab-solute value 2-D spectrum of LHCII at 77 K taken under the cross-peak-specificpolarization for T = 400 fs. The arrow highlights the Chl-b to intermediate re-gion peak obscured under the all-parallel polarization, which is visible under thecross-peak-specific polarization sequence. (c) Structural model of the chloro-phyll within monomeric LHCII, with the three chlorophyll that contribute to thepeak revealed in (b) shown in green. The site-basis transition dipole moments ofthe chlorophyll are shown (J = 0). (d) Excitonic transition dipole moments areshown (J �= 0) and the site energies are EChlb606 = 15680 cm−1 and EChlb607= 15730 cm−1 . (e) Excitonic transition dipole moments are shown with J �= 0and for a different set of site energies (EChlb606 = 15780 cm−1 and EChlb607= 15700 cm−1 ). Reprinted with permission from the Proc. Natl. Acad. Sci.USA Figure first appeared in [55].

From the dynamics of this peak (fast population and slowdepopulation, observed in the 2-D spectra of [55]), the partici-pating chlorophylls were assigned to Chl-b 605, 606, and 607(numbering from the X-ray crystallography model). Accordingto the dynamical model, the higher energy, strongly coupledChl-b 606 and 607 give rise to donor states, and Chl-b 605,a lower energy chlorophyll weakly coupled to the Chl-a band(according to models which match previous experimental re-sults [27], [28]), gives rise to the acceptor state. As explainedearlier, from the polarization dependence, the angle betweendonor and acceptor transition dipole moments must be ∼90◦.Because this angle is a function of site-basis transition dipolemoments, couplings between chlorophyll, and site-basis tran-sition energies, these parameters must produce angles betweenthe transition dipole moments of one donor and the acceptorclose to 90◦. Dependence of angles between transition dipolemoments on these parameters is illustrated in Fig. 6(c)–(e).The three chlorophylls contributing to the highlighted peak ap-pear in green (Chl-bs 605, 606, and 607). Fig. 6(c) shows thetransition dipole moments of uncoupled chlorophyll (J = 0).Fig. 6(d) shows the excitonic transition dipole moments forcoupled chlorophyll (J �= 0), and with site energies determinedthrough this technique. Fig. 6(e) shows the excitonic transitiondipole moments also in the presence of coupling, but with anarbitrary set of site energies. For the chlorophylls shown here,couplings and site-basis transition dipole moments can be esti-mated from the model from X-ray crystallography, leaving thesite energies unknown. The site energies were determined to bebetween 15 630 and 15 710 cm−1 for Chl-b 606, and 15 680 and15 760 cm−1 for Chl-b 607. These are the only values that, for



Fig. 7. (a) Diagonal slices from absolute value, nonrephasing spectra of LHCIIat 77 K under the all-parallel polarization versus waiting time. Clear oscilla-tions are visible within the Chl-a and Chl-b region, lasting through the 500 fsof the scan. (b) Fourier transform across waiting time produces a power spec-trum, where vertically aligned peaks indicate exciton energies. Reprinted withpermission from the Phys. Chem. Chem. Phys. USA Figure first appeared in [10].

these couplings and site-basis transition dipole moments, couldgive the angle between transition dipole moments observed atthe energetic coordinates of the peak.

Measuring the geometry of these delocalized excited statesalso has the potential to offer insight into the design princi-ples behind photosynthetic complexes. Delocalized states canchange the energies and transition dipole moments of the ex-cited states. This broadens the spectrum to utilize more incidentsunlight. Delocalized states can also redistribute the directionof the transition dipole moments at each absorption frequency;therefore, the photosynthetic apparatus can absorb isotropicallythrough more of the solar spectrum [55]. Finally, delocalizedstates can lead to multiple excited states on a single part of thecomplex. For example, in this case, multiple states can be lo-cated proximal to the reaction center, thus ensuring that at leastone of them is oriented favorably for energy transfer toward thereaction center [28].

V. QUANTUM COHERENCE AND ENERGY

LANDSCAPE DETERMINATION

Quantum coherences are the ensemble-averaged superposi-tion states between excitons, which, as explained earlier, arethemselves superposition states of the excitations of the indi-vidual chromophores. If undisturbed by environmental factors,each coherence |ej 〉〈ek | between excitons will evolve freelyduring the waiting time T, acquiring an oscillatory phase factore−iωj k T , where ωjk is the energy difference between the twoexcitons ej and ek . As described in Section III, quantum beat-ing from this oscillation can be detected in 2-D spectra. 2-Dspectroscopy is sensitive to the phase evolution of the signalbecause the signal electric field is detected at the amplitude, notintensity, level by interferometry [5]. As shown in Fig. 4, coher-ences appear on the diagonal in nonrephasing spectra, and thus,diagonal peak amplitudes beat as a function of waiting time T,for as long as coherences persist in the ensemble [57].

Such coherences were observed in 2-D spectra of LHCIIwith all-parallel polarization of the incident laser pulses [47].Fig. 7(c) displays the diagonal slices from the nonrephasingspectra as a function of waiting time with 5-fs steps. These

amplitude oscillations are a sign of excitonic coherence. Fur-ther evidence that enables these oscillations to be distinguishedfrom vibrational coherences is given in [44]–[46]. Quantumbeating appears through the Chl-a and Chl-b region and seemsto persists beyond 500 fs, the time scanned in the experiment.As discussed in the following section, this is longer than manyof the energy transfer times within this system. The long life-time of the quantum-beating signal is surprising because thereis a large number of degrees of freedom in the protein bath(phonon modes of the surrounding protein matrix), which cou-ple to the transitions and could destroy phase coherence. Someexperimental results, however, suggest that the fluctuations ofneighboring sites may be correlated, thus preserving the coher-ence [58]. The observations of long-lived coherence [44]–[46]have led to speculation that coherence may contribute to op-timizing photosynthetic energy transfer efficiency [59], [60].The balance of coherent and incoherent energy transfer maychange the rates of excitation energy transfer by playing a rolein unidirectionality of energy flow [11], [41], and helping ensurethat light-harvesting is robust to variations in site energies andtemperature.

In addition to providing insights into the mechanisms of en-ergy transfer, the quantum-beat signal can be used to identifythe energy landscape, or the energies of the excited states [47].Because of the overlap of closely spaced transitions, not onlythe dynamics, but also the energies of the excited states are dif-ficult to resolve. Fourier transforming the diagonal cut (ωτ =ωt) of a nonrephasing 2-D spectrum as a function of the waitingtime T [see Fig. 7(a)] produces a power spectrum as a functionof ωT [see Fig. 7(b)]. The peaks occur at the energy differencesbetween excitons (quantum-beat frequencies). A portion of thepower spectrum, the Chl-a region, is shown in Fig. 7(b). Asshown in this spectrum, the Chl-a excitons are at 14 700, 14 770,14 810, 14 880, 14 910, 14 990, 15 030, and 15 130 cm−1 . Thistechnique directly accesses the energy landscape because beatfrequency peaks align vertically at coordinates correspondingto the excited state energies. Identification of these values asdescribed here was achieved with a single experiment [47]. Theproblem was previously approached through a combination ofmany experiments and simulations [4].

VI. MONITORING PATHWAYS OF ENERGY FLOW

As described earlier, 2-D spectroscopy can be used to un-derstand excited state energies and geometry and measure co-herence of excitons. The underlying question, however, is howthese pieces combine to give rise to efficient excitation energytransfer. As discussed in Section III, conventional techniques ofmeasuring excitation energy flow integrate over excitation fre-quencies, and therefore, obscure correlation between particularexcitation and emission events. In 2-D spectroscopy, the abil-ity to resolve dynamics along excitation, emission, and waitingtimes helps in unraveling complicated dynamics. Resolution ofdynamics along excitation and emission frequencies is particu-larly critical in LHCII because the 14 chlorophyll within eachmonomer produce 14 closely spaced energy levels that give rise



Fig. 8. Real, 2-D spectra of LHCII at 77 K with relaxation (left) and non-rephasing (right) for T = 30 fs, 70 fs, 300 fs, and 13 ps. (a) Energy transferswithin all spectral regions by 30 fs, as indicated by the emergence of crosspeaks. (b) Increased relative amplitude in the midenergy Chl-a region by 70 fsindicates continued dynamics on a sub-100-fs time scale. (c) Increased relativeamplitude of the midenergy Chl-a region indicates additional energy transferprocesses occurring at ∼300 fs. (d) New cross peak (6) shows a long-livedintermediate state on a picosecond time scale. Reprinted with permission fromthe J. Phys. Chem. B. Figure first appeared in [28].

to overlapping features and a large range of energy transfertimes [28].

Fig. 8 shows the real part (absorptive component) of relax-ation and nonrephasing 2-D spectra for LHCII taken under theall-parallel polarizations. In Fig. 8(a), by 30-fs energy transferhas occurred among all spectral regions, as indicated by the ap-pearance of the five crosss-peaks highlighted by arrows in thenonrephasing spectrum. By 70-fs [see Fig. 8(b)], the relative am-plitude of these peaks has changed, with increased amplitude inthe low-energy Chl-a band (cross peaks 1, 2, and 4). These crosspeaks indicate a complex pattern of energy transfer between allspectral regions within the first 100 fs. The appearance, andthen evolution of the relative amplitude of these peaks, signi-fies that multiple energy transfer steps occur on this ultrafasttime scale. On a several hundred femtosecond time scale, an-other set of energy transfer steps occurs. As shown in Fig. 8(c),there is an increase in relative amplitude of peaks 3 and 5 by300 fs, which then decrease after a picosecond. These dynamicsshow that energy transfers into the midenergy Chl-a region ona few hundred femtosecond time scale, and then relaxes downto the low-energy Chl-a band after an additional few hundred

femtoseconds. Multiple energy transfer steps on the same timescale as that of coherence open the possibility that coherencecould play a role in honing the energy transfer steps.

On a picosecond time scale, a sixth cross peak appears thatindicates energy transfer from Chl-b into the high-energy Chl-aband. The peak appears at around 1 ps and population remainsin this state for the lifetime of the signal (20 ps). The cross peakobserved in the cross-peak-specific data also shows populationremaining in an intermediate state for the lifetime of the signal(5 ps).

Overall, there is a large range of time scales of energytransfer steps within LHCII, which combine to aid in unidi-rectional energy flow. The fast energy transfer rates, drivenby strong coupling, can aid in relaxation across large en-ergy gaps and prevent thermal energy from facilitating trans-fer back to a previous state. The weaker coupling, whichis responsible for the slower energy transfer steps, doesnot ensure long-lived superpositions between coupled states.This helps in excitation transport by preventing continuoussampling of states within a local region. The interplay be-tween processes on multiple time scales has been observedin other photosynthetic complexes as well, such as the FMOcomplex [41].

VII. CONCLUSION AND OUTLOOK

In this paper, we have reviewed the principles behind mul-tidimensional spectroscopy and presented several applicationsof this technique to photosynthetic light-harvesting systems.2-D spectroscopy is a powerful method to elucidate structureand dynamics in photosynthetic complexes. For the most abun-dant light-harvesting protein of plants LHCII, this technique canprovide a picture of the energy transfer dynamics, and how themolecular structure produces the observed behavior. In combi-nation with the modeling based on the X-ray crystallographydata, and aided by structural information provided by polar-ization experiments, an understanding of energy flow throughthe complex emerges. Information from coherence finds theenergies of the excited states and gives information about themechanisms by which energy is transferred through the com-plex. By comparing time scales of energy transfer and the os-cillatory quantum coherence signals, the mechanisms by whichphotosynthetic energy transfer occurs can be elucidated. Asa technique, 2-D spectroscopy provides important insight intobiological function by accessing the electronic structure, the en-ergy landscape, and energy transfer time scales. While we havelimited our discussion to LHCII, 2-D spectroscopy has alreadybeen used to study the dynamics of electronic and vibrationaltransitions in many other biological systems [8], [44], [61]–[64]

2-D spectroscopy is the optical analog of multidimensionalnuclear magnetic resonance. Over the decades since the inven-tion of multidimensional NMR, an arsenal of pulse sequenceshas been constructed to target highly specific aspects of sam-ple structure and dynamics [66], often with experimental facil-ity. It is expected that a similar level of development can beachieved in the optical regime. In fact, many groups are explor-ing different techniques to streamline the measurements through



using various beam geometries [34], [38], [65], [67], phase re-trieval methodologies [68], [69], and assessing and removal ofartifacts [70]. Additionally, besides the polarization sequencesdescribed here, a variety of pulse sequences, with different tim-ings, frequencies, and polarizations, are already being proposed,modeled, and implemented [71], [72]. These targeted probesoffer great potential to disentangle the complex dynamics ofmany biological systems.

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Gabriela S. Schlau-Cohen received the Sc.B. degree in chemical physics fromBrown University, Providence, Rhode Island, in 2003. She is currently workingtoward the Ph.D. degree in physical chemistry at Lawrence Berkeley NationalLaboratory and the University of California, Berkeley.

Her current research interests include using nonlinear spectroscopy to studyphotosynthetic energy transfer, and structure–function relationships of light-harvesting complexes.

Jahan M. Dawlaty was born in Kabul, Afghanistan, in 1976. He received theB.A. degree in chemistry from Concordia College, Moorhead, MN, in 2001,and the Ph.D. degree in physical chemistry from Cornell University, Ithaca, NY,in 2008.

He is currently a Postdoctoral Fellow in the Chemistry Department, Uni-versity of California, Berkeley, and is supported by the California Institute ofQuantitative Biosciences QB3. His research interests include the method devel-opment and applications of nonlinear optical spectroscopy for understandingmolecular dynamics of energy and charge transport.

Graham R. Fleming received the B.S. degree from the University of Bristol in1971, and the Ph.D. degree in chemistry under the direction of George Porterfrom the University College London/The Royal Institution in 1974.

He did postdoctoral work with Wilse Robinson and joined the University ofChicago in 1979. In 1997, he moved to the University of California, Berkeleyand Lawrence Berkeley National Laboratory. He is currently Vice Chancellorfor Research and Melvin Calvin Distinguished Professor of Chemistry at UCBerkeley. His research interests are in ultrafast processes, the primary stepsin photosynthesis and their regulation, quantum dynamics, and the electronicstructure and dynamics of nanomaterials.

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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ... Schlau-Cohen.pdfDigital Object Identiﬁer 10.1109/JSTQE.2011.2112640 spectroscopy has proved to be a valuable tool to examine the dynamics