44 July 2013 Physics Today www.physicstoday.org

OpticalmultidimensionalcoherentSPECTROSCOPY

Steven T. Cundiff and Shaul Mukamel

Techniques developed decades ago for nuclearmagnetic resonance and now adapted for the

IR, visible, and UV regions of the spectrum, are enabling new insights into chemical

kinetics and solid-state physics.

Optical spectroscopists strive to furtherour understanding of matter by study-ing how it absorbs and emits light.Starting from Isaac Newton’s use of aprism to observe the spectrum of sun-

light, spectroscopic techniques remained fundamen-tally unchanged for centuries. Light was dispersed—for example, by a prism or diffraction grating—andthe intensity measured as a function of wavelength.Variation in the spectral intensity is either inherent inthe source or due to the light passing through amedium that absorbed at specific wavelengths. Thefrequency of an optical absorption or emission waseventually understood as the energy difference be-tween electronic or vibrational quantum states.

At first, spectroscopic measurements were alldone in the linear regime, in which the measured sig-nals are proportional to the intensity of the incidentlight. A new era of optical spectroscopy emergedwith the advent of the laser, which can produce lightthat is no longer weak compared to the internalfields of an atom or molecule.1 In that nonlinearregime, signals are proportional to a higher powerof the laser intensity, and moreover, two laser beamscan interact. For example, one strong “pump” beamcan saturate an absorption in a sample and thus in-crease the transmission of a weaker “probe” beam.Alternatively, wavemixing in the sample can gener-ate a “signal” beam with an entirely new directionand frequency.

It is also possible to make measurements as afunction not of frequency but of time. Time- domainspectroscopy may be likened to the use of a strobo-scope, in that short flashes of light can create stop-action images of a dynamical process such as achemical reaction or carriers relaxing in a solid.Measurements made in the time domain using laserpulses can be Fourier transformed into frequency- domain spectra. Often, the spectra are functions ofmultiple time delays and thus are multidimen-sional. (This article focuses on two-dimensionalspectroscopy, but higher-dimensional measure-ments are also possible.) Multidimensional Fourier- transform methods were developed for nuclearmagnetic resonance (NMR) spectroscopy2 in the1970s, and they are now transforming the field of ul-trafast laser spectroscopy.

Two-dimensional spectroscopyAny spectroscopic measurement that is plotted as afunction of two variables is, in a sense, a 2D spectrum.But the term “2D spectroscopy” is typically reservedfor the specific situation described in the box on page46. The electric field of a coherently generated signalis recorded over many runs as a function of two timevariables: the delay τ between excitation pulses andthe time t over which the signal is emitted. Numeri-cally taking the Fourier transform with respect toboth time variables generates a 2D spectrum as afunction of two optical frequencies—an absorptionfrequency (the transform of τ) and an emission fre-quency (the transform of t). That 2D spectrum con-

tains all the information of any 1D spectrum. For ex-ample, integrating the 2D spectrum over the emis-sion frequency gives the sample’s 1D absorptionspectrum and vice versa; more sophisticated 1D spec-tra can also be generated. But the 2D spectrum alsocontains information not present in any 1D spectrum.

The appeal of 2D spectroscopy lies largely in itsability to dissect ensembles. In the course of time,each member of an ensemble—an atom, molecule,nanostructure, or other absorber—undergoes sto-chastic evolution, and its properties fluctuate in un-controllable ways due to coupling with its environ-ment. One-dimensional measurements yield theensemble average over those stochastic trajectories.But many possible microscopic models, with differ-ent types of trajectories, could yield the same ensem-ble average. Multidimensional spectroscopy probesthe members of the ensemble at multiple time points,so it provides more information about their individ-ual trajectories than a 1D spectrum can. A similargoal can be achieved by measuring single particlesand isolating their trajectories one at a time, but thatapproach typically requires much greater effort.

A 2D spectrum can be used to determine if tworesonances are coupled or uncoupled. Coupled res-onances might occur if some species in the samplehas two transitions that share a common state: Exci-tation of one resonance can directly influence excita-tion of the other. In contrast, uncoupled resonancesmight indicate that the sample is a mixture of twospecies. A 1D spectrum cannot distinguish betweencoupled and uncoupled resonances. But in a 2Dspectrum such as the one in figure 1b (discussed inmore detail below), coupled resonances produce off-diagonal cross peaks, with the absorption frequencyof one resonance and the emission frequency of theother, whereas uncoupled resonances do not.

Another important capability of 2D spectroscopyis its ability to overcome inhomogeneous broaden-ing. In any spectrum, resonances appear not as per-fectly thin lines but as peaks of nonzero width. Partof that width is due to homogeneous broadening,which results from fundamental processes that af-fect every member of the ensemble: spontaneousreradiation of light, collisions between atoms, in-teractions with lattice vibrations in a solid, and so forth. But part may be due to inhomogeneousbroadening, which results from differences amongmembers of the ensemble, such as Doppler shiftingin a gas or structural disorder in a nanostructure.In 2D spectroscopy, homogeneous broadening isakin to coupling of resonances: The same memberof the ensemble can be responsible for slightly dif-ferent absorption and emission frequencies, so sig-nal appears off the spectrum’s diagonal. Inhomo-geneous broadening, like uncoupled resonances,

www.physicstoday.org July 2013 Physics Today 45

Steven Cundiff is a fellow of JILA and an adjoint professor of physicsand electrical and computer engineering at the University of ColoradoBoulder. Shaul Mukamel is a professor of chemistry at the Universityof California, Irvine.

produces no off-diagonal signal. In the limit of stronginhomogeneous broadening, a peak’s inhomoge-neous width is given by the width measured alongthe diagonal, and the homogeneous width may bemeasured perpendicular to the diagonal.

In principle, 2D optical spectroscopy is muchlike the established technique of 2D NMR. But im-plementing it poses many additional challenges, be-cause the frequencies are much higher and the timescales much shorter. The first challenge is measuringthe electric field of the signal. That is easily done atthe radio frequencies used in NMR, but optical de-tectors measure only the intensity. To measure an op-tical signal’s electric field, the signal must be inter-fered with a well- characterized reference pulse. Thesecond challenge is precisely timing the pulses, withdelays accurate to a fraction of an optical cycle. Thedelays must be stable while a measurement is beingmade, and the delay steps must be highly uniform.

The idea of implementing 2D coherent spec-troscopy in the optical regime was first proposed in 1993 by Yoshitaka Tanimura and one of us(Mukamel).3 That proposal described using five-pulsesequences to excite molecular vibrations through theRaman effect. Attempts to implement it experimen-tally were initially complicated by cascading effects(a sequence of lower-order signals that has the samepower dependence and direction as the desired sig-nal, making it hard to distinguish between the two).In 2002, after almost a decade, the complications wereindependently overcome by the groups of DwayneMiller and Graham Fleming. In the meantime, 2Dspectroscopy in the IR, which accesses molecular vi-brations directly, was demonstrated independentlyby the groups of Robin Hochstrasser in 2000 and An-drei Tokmakoff in 2001. Working in the IR poses chal-lenges in terms of sources and detectors, but it hasthe advantage that a well- designed apparatus usu-ally has sufficient passive stability because of thelonger wavelength. Using light in the near-IR or vis-ible parts of the optical spectrum makes it possible toaccess electronic transitions in atoms, molecules, orsemiconductors. At those wavelengths, more elabo-rate approaches are needed to fulfill the require-ments of 2D spectroscopy.4,5

Atomic vaporsThe spectroscopy of isolated atoms is well under-stood, so an atomic vapor provides an ideal illustra-tive example of the capabilities of 2D spectroscopy.Figure 1 shows the 1D and 2D spectra of the D1 andD2 transitions of potassium vapor.6 The lines corre-spond to transitions of the single outer electron ofthe potassium atom from the ground state to two dif-ferent excited states; the resonances are thereforecoupled. As a result, the 2D spectrum shows fourpeaks. Two peaks, labeled A and B, lie on the diago-nal and correspond to the D1 and D2 transitions beingdriven independently. Peaks C and D are off-diago-nal cross peaks, with the absorption frequency of onetransition and the emission frequency of the other,as expected for coupled resonances. Two physicalprocesses contribute to the cross peaks. First, drivingone transition reduces the absorption in the other be-cause it reduces the number of atoms available to be

46 July 2013 Physics Today www.physicstoday.org

Spectroscopy

Optical two-dimensional spectroscopy uses a sequence of laser pulsesthat are incident on the sample from different directions. A typicalgeometry is shown in the top panel of the figure. The incident pulses, A,B, and C, generate a signal S that emerges in a different direction, deter-mined by conservation of momentum, and thus is easily isolated.

As shown in the bottom panel, the pulses are separated by time delaysτ and T, and the signal is emitted over time t. The first pulse, with wavevec-tor kA, puts the system in the coherent superposition between the

ground state ∣0⟩ and the excited state ∣1⟩. The superposition oscillates asa function of time, as indicated by the decaying sinusoid. The secondpulse, kB, converts the superposition to a population in the excited state.The excited-state population does not oscillate, but it does store thephase of the superposition state. The third pulse, kC, converts the popula-tion back to a coherent superposition state that radiates the signal fieldwith wavevector kS = −kA + kB + kC. Pulse A contributes a minus sign be-cause its contribution to the phase of the signal is conjugated, or shiftedby 180° with respect to the contributions of the other two pulses. A 2Dspectrum is then generated by scanning τ, recording the signal as a func-tion of t, and taking Fourier transforms with respect to τ and t.

That pulse sequence, with the phase-conjugated pulse arriving first,produces a so-called rephasing spectrum, which is only one of severalpossible 2D spectra. If the conjugated pulse arrives second, the resultingspectrum is known as a “nonrephasing” spectrum because the dephas-ing due to inhomogeneous broadening is not canceled, or “rephased.” Asshown by Andrei Tokmakoff, if rephasing and nonrephasing spectra areadded together, the resulting “correlation” spectrum has narrower peaks.Correlation spectra can disentangle congested spectra because theyhave narrower features.

If the conjugated pulse arrives last, the resulting spectrum can show double- quantum coherences, or coherent superpositions of the groundstate and a doubly excited state. Such coherences cannot be directly cre-ated by a single light pulse because the transition between these statesis not dipole allowed. However, two non-conjugated pulses together cancreate a double- quantum coherence, which oscillates at the frequencycorresponding to the energy difference between the ground state andthe doubly excited state, typically around twice that of the single quan-tum transition.

Pulse and excitation sequence

AB

C

S

kA∗

kA∗

kB

kB

kC

kC

Time

T tτ

∣1⟩

∣0⟩

Sample

excited. Second, atoms can be put in a coherent su-perposition of the two excited states. In other sys-tems, but not potassium vapor, cross peaks can occurdue to decay from one excited state to another.

As described in the box, changing the order ofthe excitation pulses generates several different typesof spectra. Figure 2 shows the double-quantum spec-trum of potassium in the spectral region correspon-ding to twice the frequency of the D1 and D2 transi-tions.7 Based on the level structure of an isolatedpotassium atom, which has no excited states at thoseenergies, there should be no features in that spectralrange. The double- quantum coherences are due to theinteraction of two atoms. The double quantum coher-ence occurs between the state in which both atoms areunexcited and the state in which both are excited.

MoleculesMolecules present much richer spectra than atoms,in part because of nuclear motion. Like a system ofinterconnected balls and springs, a polyatomic mol-ecule can be excited into a range of vibrational mo-tions. Figure 3 shows a 2D spectrum of dicarbony-lacetylacetonato rhodium, one of the first moleculesto be studied with 2D vibrational spectroscopy.8 Thediagonal peaks at 2015 and 2084 cm−1 correspond tothe asymmetric and symmetric stretching modes ofthe molecule’s two carbonyl, or CO, groups. Thosemodes are coupled, so the spectrum contains crosspeaks at the same frequencies. If the vibrationalmodes were perfect harmonic oscillators, withequally spaced quantum states, those would be the

only four peaks in the spectrum. But molecular vi-brations are anharmonic: The energy of a second vi-brational quantum is less than the energy of the first.Those energy differences are reflected in the offsetsof the peaks shown in shades of blue. The anhar-monicity cannot be determined from the 1D linearabsorption spectrum, because peaks at the relevantenergies are so closely spaced that they would beimpossible to distinguish.

An important advantage of using short pulsesis the ability to probe reaction dynamics. Considera chemically reactive system in which molecularspecies A and B can interconvert. If the system is atequilibrium and the reaction rates are slow com-pared with the spectroscopic frequencies (corre-sponding to times t and τ as defined in the box), the1D absorption spectrum is simply the weighted av-erage of the spectra of A and B; no informationabout the reaction kinetics is available. In nonlinear2D spectroscopy, one can vary the time delay T onthe kinetic time scale, which can be much longerthan τ, and extract the kinetics from the time evolu-tion of the cross peaks. In 2005 the group of MichaelFayer used 2D IR spectroscopy to study the forma-tion and dissociation of solute–solvent complexes.9

As shown in figure 4, phenol (the solute) forms hy-drogen-bonded complexes with benzene, the sol-vent. The complexes are now known to have life-times on the order of 10 ps. When T is much lessthan 10 ps, the vibrational resonances of the com-plex and those of the free phenol are uncoupled, sothey produce no cross peaks. When T is greater than10 ps, cross peaks appear. The variation of cross-peak volume with delay time measures the creationand dissociation rates. Such picosecond chemicalexchange processes can be probed with 2D IR spec-troscopy, but not with 2D NMR, because the t and τrequired for NMR are in the milliseconds.

www.physicstoday.org July 2013 Physics Today 47

2D1

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Figure 2. A double-quantum spectrum.The same potassiumvapor used in figure 1yields a very differentspectrum when theorder of the excitationpulses is changed.(Adapted from ref. 7.)

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Figure 1. Potassium vapor is a simple three-level system for demonstrating two-dimensionalspectroscopy. The 1D absorption spectrum (a) shows two peaks, D1 and D2, which resultfrom electronic excitation from the ground stateinto two different excited states. (The red curve isthe laser-pulse spectrum, which is broad enoughto excite both transitions.) In the 2D spectrum (b), off-diagonal cross peaks C and D indicate that the resonances are coupled. (The absorptionfrequency here is negative by convention.)

Hydrogen bonds are weak chemical bonds (witha binding energy on the order of the thermal energyat room temperature) involving hydrogen and an ac-cepting atom with high electronegativity—usuallyoxygen, nitrogen, or fluorine. They are responsible forthe tetrahedral structure of water ice and many ofwater’s important and anomalous properties. Theyare crucial to the structure of proteins and the double-helix structure of DNA. Their relative ease of formingand breaking makes them important in many biolog-ical processes such as enzymatic catalysis. At the sametime, because they are so weak, they tend to form avariety of interconverting structures, which yieldbroad IR spectral peaks that are hard to interpret.

Multidimensional spectroscopy can help to dis-sect those complex spectra. Several research groupshave used 2D spectra of liquid water to probe thetime scales on which hydrogen bonds form andbreak and to explore the connectivity of different hy-drogen-bonded clusters. Some researchers, such asTokmakoff and Fayer, simplified the spectra by usingisotopically labeled samples—for example, HDO inH2O. Others, including Miller and Thomas Elsässer,used pure H2O. The groups of Hochstrasser, PeterHamm, and Martin Zanni have used the same tech-nique to study hydrogen bonding in proteins.

Multidimensional coherent spectroscopy ofelectronic transitions has been used extensively inthe past few years by the groups of Fleming andGregory Scholes to study charge and energy trans-port in photosynthetic light- harvesting complexes.The dependence of the cross-peak pattern on thewaiting time T provides detailed snapshots of theentire relaxation pathway among the variouschlorophyll molecules. (See PHYSICS TODAY, July2005, page 23.)

SemiconductorsThe nonlinear optical response of semiconductorsreflects the dynamics of electron–hole pairs createdby the excitation of an electron to the conductionband from the valence band. The excited electroncan bind to the hole to form an exciton, which has ahydrogenic wavefunction in the relative coordinate.Because the excitonic binding energy is so weak—approximately three orders of magnitude weakerthan for a hydrogen atom—excitons exist only attemperatures around 10 K. Despite being neutralparticles that do not exhibit direct Coulomb interac-tions, excitons display strong many- body effects.10

As a result, excitons in semiconductors are an idealsystem for studying fundamental many- bodyphysics, which in turn is critical for developing afirst- principles understanding of optoelectronic de-vices such as laser diodes.

Direct-bandgap materials, such as gallium ar-senide and related compounds, are typically used inoptoelectronic applications because of their strongabsorption and emission of light. Often, the opticallyactive layers are made thin enough that the motionof the charge carriers is quantized in the directionperpendicular to the plane of the layer, thus forminga quantum well. In bulk GaAs, different hole spinstates are known as heavy holes and light holes dueto their different effective masses. In a quantum well,their energies are split by quantum confinement.

Excitons in quantum wells have been studied extensively using traditional 1D ultrafast spectro-scopies. But 1D techniques are limited in their abilityto probe the nonlinear optical response that arisesfrom many-body effects. Because 2D spectra, such asthe one shown in figure 5, measure not just the mag-nitude of the coherent signal but also its real andimaginary parts (the phase of the signal with respectto the phases of the excitation pulses), they make itpossible to determine the microscopic many- bodyphenomena that underlie the nonlinear optical re-sponse.11 Specifically, the diagonal heavy-hole peakshows a dispersive lineshape—that is, in the cross-

48 July 2013 Physics Today www.physicstoday.org

Spectroscopy

Figure 4. Phenol dissolved in benzene forms hydrogen-bonded com-plexes (a) with a lifetime τd of 10 ps, as measured by two-dimensionalspectroscopy. Carbon atoms are shown in gray, hydrogen in white, and

oxygen in red. (b) Whenthe delay time T (as de-fined in the box on page46) is much less than τd,vibrational resonances ofthe complex and the free phenol produce no crosspeaks. When τd > T, crosspeaks appear. (Adaptedfrom ref. 9.)

SYMMETRICASYMMETRIC

Δas

Δas

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Δs

2000 2045

2090

2045

2090

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ABSORPTION (cm )−1

EM

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ION

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)−1

O

O

O

OC

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Rh

Figure 3. A 2D IR spectrumof dicarbonylacetylacetonatorhodium (top) showing thesymmetric and asymmetricstretching modes of the car-bonyl (CO) groups. The fourpeaks shown in shades ofred represent single excita-tions of the molecule fromits vibrational ground state.The emission energies of thepeaks shown in shades ofblue represent the energyrequired to excite a mole-cule that already has onequantum of vibrational en-ergy. The energy shifts Δa, Δs,and Δas quantify the modes’anharmonicity—that is, howmuch they deviate fromideal harmonic oscillators.(Adapted from ref. 8.)

diagonal direction it goes positive and negative witha node on the diagonal—that can only be reproducedtheoretically if many- body effects are considered.

A detailed comparison of 2D spectra to theoret-ical results definitively showed that a simple mean-field theory does not suffice to describe the excitonsystem.12 In a mean-field approximation, each exci-ton is assumed to interact with an average of allother excitons. Higher-order correlations describeinteractions between two excitons, including ex-plicit bound states, or biexcitons, as well as pairwiseinteractions that do not form bound states. Al-though the importance of the higher-order termswas evident from theoretical considerations, it wasdifficult to show that they were required based on1D measurements, which could often be accuratelyreproduced by simpler theories.

Semiconductor quantum dots grown using epi-taxial methods have attracted significant attention fortheir possible applications in quantum informationscience. (See the article by Dan Gammon and DuncanSteel, PHYSICS TODAY, October 2002, page 36.) Becausetheir charge-carrier wavefunctions are confined in allthree dimensions, quantum dots, like atoms, havediscrete energy levels. The energy levels can be engi-neered to some extent by adjusting the size of the dot,but it is difficult to make an ensemble of quantumdots that are all exactly the same size. Most studies,therefore, have been performed on single dots; thoseexperiments are very challenging and make it diffi-cult to accumulate enough measurements to accu-rately determine trends. However, 2D spectroscopycan make size-resolved measurements on an inho-mogeneous distribution of dots. In addition, because2D spectroscopy excels in detecting and characteriz-ing coupling between resonances, Wolfgang Lang-bein and coworkers were able to show that natural

quantum dots, which form due to thickness fluctua-tions in thin quantum wells, can couple to one an-other.13 Measuring the coupling strength betweenquantum dots is critical to quantum information ap-plications that require controllable coupling.

OutlookThe field of multidimensional coherent spectroscopyis still evolving, and new developments are occur-ring frequently. One interesting future direction isto measure incoherent observables rather than a co-herently generated optical signal. Coherence infor-mation is nevertheless obtained through the para-metric dependence of the incoherent signals ontime delays. For example, a fluorescence signal wasused in the early work of Warren Warren’s groupand has also been exploited by Andrew Marcus. Inmuch more challenging experiments, the groups ofTobias Brixner and Martin Aeschlimann combinedmultidimensional coherent methods with a photoe-mission electron microscope to construct a 2D spec-trum of a corrugated silver surface as a function ofspatial position, with sub- wavelength resolution.Their measurements showed that plasmonic phasecoherence of localized modes can persist for a sur-prisingly long 100 fs.

Another direction is to extend 2D spectroscopicmethods to higher frequencies. Two- dimensionalcoherent spectroscopy in the UV was recentlydemonstrated by the groups of Thomas Weinachtand Andrew Moran. New x-ray sources, includingfree-electron lasers and tabletop high-harmonicsources, may enable resonant attosecond nonlinearultrafast x-ray spectroscopies of core excitons inmolecules. X-ray pulses have the unique capacity todetect nuclear and electronic motions with both at-tosecond and nanometer resolution. New methodsfor looking at electron–electron correlations and theflow of charge are possible. X-ray pulses can also directly probe structural fluctuations, relaxationprocesses, and the dynamics of elementary excita-tions such as excitons, polaritons, and polarons inreal time and space.

References1. S. Mukamel, Principles of Nonlinear Optical Spec-

troscopy, Oxford U. Press, New York (1995).2. R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of

Nuclear Magnetic Resonance in One and Two Dimensions,Oxford U. Press, New York (1987).

3. Y. Tanimura, S. Mukamel, J. Chem. Phys. 99, 9496 (1993).4. S. Mukamel, Annu. Rev. Phys. Chem. 51, 691 (2000);

D. Jonas, Annu. Rev. Phys. Chem. 54, 425 (2003); M. Cho, Chem. Rev. 108, 1331 (2008); D. Abramaviciuset al., Chem. Rev. 109, 2350 (2009).

5. P. Hamm, M. Zanni, Concepts and Methods of 2D In-frared Spectroscopy, Cambridge U. Press, New York(2011).

6. X. Dai et al., Phys. Rev. A 82, 052503 (2010).7. X. Dai et al., Phys. Rev. Lett. 108, 193201 (2012).8. M. Khalil, N. Demirdöven, A. Tokmakoff, Phys. Rev.

Lett. 90, 047401 (2003); J. Chem. Phys. 121, 362 (2004).9. J. Zheng et al., Science 309, 1338 (2005).

10. D. S. Chemla, J. Shah, Nature 411, 549 (2001).11. S. T. Cundiff et al., Acc. Chem. Res. 42, 1423 (2009).12. T. Zhang et al., Proc. Natl. Acad. Sci. USA 104, 14227

(2007).13. J. Kasprzak et al., Nat. Photonics 5, 57 (2011). ■

www.physicstoday.org July 2013 Physics Today 49

AB

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N E

NE

RG

Y(m

eV)

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CE

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EMISSION ENERGY meV)(

1

0

−1

a

b

Heavyhole Light

hole

Figure 5. In a gallium arsenide quantum well, theexciton resonance is split into two peaks (a) thatcorrespond to holes with different effective masses.(The laser-pulse spectrum is shown in red.) In thetwo-dimensional spectrum (b), the intricate peakshapes arise from the many-body physics of exciton–exciton interactions. (Adapted from ref. 11.)