Chemical Physics Letters 545 (2012) 40–43

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Chemical Physics Letters

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Vibrational vs. electronic coherences in 2D spectrum of molecular systems

Vytautas Butkus a,b, Donatas Zigmantas c, Leonas Valkunas a,b,⇑, Darius Abramavicius a,d

a Department of Theoretical Physics, Faculty of Physics, Vilnius University, Sauletekio 9-III, LT-10222 Vilnius, Lithuaniab Center for Physical Sciences and Technology, Gostauto 9, LT-01108 Vilnius, Lithuaniac Department of Chemical Physics, Lund University, P.O. Box 124, 22100 Lund, Swedend State Key Laboratory of Supramolecular Complexes, Jilin University, 2699 Qianjin Street, Changchun 130012, PR China

a r t i c l e i n f o

Article history:Received 23 May 2012In final form 5 July 2012Available online 15 July 2012

0009-2614/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.cplett.2012.07.014

⇑ Corresponding author at: Department of TheoreticVilnius University, Sauletekio 9-III, LT-10222 Vilnius,

E-mail address: leonas.valkunas@ff.vu.lt (L. Valkun

a b s t r a c t

Two-dimensional spectroscopy has recently revealed the oscillatory behavior of the excitation dynamicsof molecular systems. However, in the majority of cases there is considerable debate over what is actuallybeing observed: excitonic or vibrational wavepacket motion or evidence of quantum transport. In this let-ter we present a method for distinguishing between vibrational and excitonic wavepacket motion, basedon the phase and amplitude relationships of oscillations of distinct peaks as revealed through a funda-mental analysis of the two-dimensional spectra of two representative systems.

� 2012 Elsevier B.V. All rights reserved.

Two-dimensional photon-echo (2DPE) spectroscopy is a power-ful tool capable of resolving quantum correlations on the femtosec-ond timescale [1–3]. They appear as beats of specific peaks in the2DPE spectrum for a number of molecular systems [3,4]. However,the underlying processes are often ambiguous. At first, the beatswere attributed to the wave-like quantum transport with quantumcoherences being responsible for an ultra-efficient excitationtransfer [3–6]. The same process was associated with the oppositephase beats in the spectral regions which are symmetric withrespect to the diagonal line [7].

In molecules and their aggregates, electronic transitions arecoupled to various intra- and intermolecular vibrational modes.Vibrational energies of these are of the order of 100–3000 cm�1,while the magnitudes of the resonant couplings, J, in excitonicaggregates (e.g. in photosynthetic pigment-protein complexes orin J-aggregates) are in the same range. Thus, vibronic and excitonicsystems show considerable spectroscopic similarities, and pres-ence of electronic and/or vibrational beats in the 2DPE spectrumis expected. Indeed, similar spectral beats originating entirely froma high-energy vibrational wavepacket motion have been studiedtheoretically for weak electron–phonon coupling [8] and have beenobserved experimentally [9]. The possibility of distinguishing theelectronic and vibrational origin of the beats from a 2DPE spectrumhas been emphasized in a recent letter [10]. However, the reportedconclusions have not been supported by theoretical arguments,and thus are questionable. Therefore, the highly relevant questionof how vibrations interfere with electronic coherences in 2DPEspectrum is still an open one. A theoretical study of the origin ofspectral beats, their phase relationships in the rephasing and

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al Physics, Faculty of Physics,Lithuania.as).

non-rephasing components of the 2DPE spectrum is presented inthis article.

We address this problem by considering two generic model sys-tems which exhibit distinct internal coherent dynamics. The sim-plest model of an isolated molecular electronic excitation is thevibronic system represented by two electronic states, jgi and jei,which are coupled to a one-dimensional nuclear coordinate q.We denote the model by a displaced oscillator (DO) system (Figure1a). Taking �h ¼ 1, the vibronic potential energy surface of the jeistate is shifted up by electronic transition energy xeg and its min-imum is shifted by d with respect to the ground state jgi; d is thedimensionless displacement. This setup results in two vibrationalladders of quantum sub-states jgmi and jeni, m;n ¼ 0 . . .1, charac-terized by the Huang–Rhys (HR) factor HR ¼ d2

=2 [11,12].The other model system, which shows similar spectroscopic

properties but has completely different coherent internal dynamicswithout vibrations, is an excitonic dimer (ED). It consists of twotwo-level chromophores (sites) with identical transition energies�. The two sites are coupled by the inter-site resonance couplingJ. As a result, the ED has one ground state jgi, two single-excitonstates je1i and je2i with energies ee1 ;e2 ¼ �� J, respectively, and asingle double-exciton state jf i with energy ef ¼ 2�� D, where Dis the bi-exciton binding energy (Figure 1b) [13].

The absorption spectrum of both systems is as follows. Theabsorption of the DO is determined by transitions from the jgmivibrational ladder into jeni scaled by the Franck–Condon (FC) vibra-tional wavefunction overlaps [11,13]. Choosing HR ¼ 0:3 andkBT � 1

3 x0 and assuming Lorentzian lineshapes with linewidth c,we get the vibrational progression in the absorption spectrum(dashed line in Figure 1). Here x0 is the vibrational energy. Themost intensive peaks at xeg and xeg þx0 correspond to 0–0 and0–1 vibronic transitions. Qualitatively similar peak structure isfeatured in the absorption of ED, where the spectrum shows two

a bFigure 1. Energy level structure of the displaced oscillator (a) and electronic dimer(b) and corresponding linear absorption spectra.

Figure 2. (a) Scheme of contributions to 2DPE spectrum of the kI and kII signals forthe reduced DO and ED (D ¼ 0) systems. The ESE contribution is indicated bysquares, GSB – diamonds, ESA – circles. Solid symbols denote non-oscillatingcontributions in t2, open – oscillatory in the form of � cosðe2t2Þ, where e2 ¼ x0 forDO and e2 ¼ 2J for ED. (b) Phase / of the contribution (Eq. 3) and peak profile Sn as afunction of the shift from the peak center (s1 ¼ s3 ¼ 0) using relative coordinatesThe diagonal lines of the kI and kII contributions to the 2D spectra are shown bydashed lines. Peaks are labeled in plots as ‘1-1’, ‘1-2’, etc.

V. Butkus et al. / Chemical Physics Letters 545 (2012) 40–43 41

optical transitions jgi ! je1i and jgi ! je2i, assuming both are al-lowed. Choosing J ¼ x0=2 and the angle u between the chromo-phore transition dipoles equal to p=6, and using adequatelinewidth parameters, we get absorption peaks (solid line in Figure1) that exactly match the strongest peaks of the DO. As expectedone cannot distinguish between these two internally different sys-tems from the absorption spectra alone.

The 2DPE spectrum carries more information than absorption.However, it consists of many contributions and unambiguousassignment of peaks and their dynamics becomes difficult. In orderto unravel the 2DPE spectra we thus need to construct the entire2D signal from the first principles for both systems and recoverthe source of oscillations in the 2DPE spectrum.

In the conventional scheme of the 2DPE measurement, two pri-mary excitation pulses with wavevectors k1 and k2 followed by theprobe pulse k3 are used; kj are pulse wavevectors. The signal is de-tected at the kS ¼ �k1 þ k2 þ k3 phase-matching direction. The or-der of k1 and k2 defines the rephasing configuration (kI) when k1

comes first and the non-rephasing configuration (kII) when k2

comes first.Semiclassical perturbation theory with respect to the incoming

fields reveals the system-field interaction and evolution sequences,often denoted by the Liouville space pathways. Three types of dis-tinct interaction configurations are denoted by the Excited StateEmission (ESE), Ground State Bleaching (GSB) and Excited StateAbsorption (ESA) [12,14]. If we neglect environment-inducedrelaxation, the signals are given as sums of resonant contributions,Sðx3; t2;x1Þ ¼

PnSnðx3; t2;x1Þ of the type

Snðx3; t2;x1Þ ¼ AðnÞ

ZZdt1dt3eþix3t3þix1t1

� ½�G3ðt3ÞG2ðt2ÞG1ðt1Þ�ðnÞ; ð1Þ

where the subscript n denotes different terms of the summation.AðnÞ is a complex prefactor, given by the transition dipoles and exci-tation fields, the propagator of the density matrix G for the jth(j ¼ 1;2;3Þ time delay is of the one-sided exponential function type

GjðtjÞ ¼ hðtjÞ expð�iejtjÞ ð2Þ

(hðtÞ is the Heaviside step-function). Here ej coincides with the en-ergy gap xab between the left and right states of the system densitymatrix relevant to the time interval tj. ESE and GSB carry ‘+’ signwhile ESA has ‘�’ overall sign.

The Fourier transforms in Eq. (1) map the contributions to thefrequency-frequency plot ðt1; t3Þ ! ðx1;x3Þ � ð�je1j; e3Þ (theupper sign is for kI, the lower – for kII). Diagonal peaks at x1 ¼�x3 are usually distinguished, while the anti-diagonal line is de-fined as �x1 þx3 ¼ Const. The whole 2DPE signal becomes a func-tion of t2: either oscillatory for density matrix coherences jaihbj

with characteristic oscillation energy e2 ¼ xab – 0, or static forpopulations jaihaj (e2 ¼ 0).

To reveal oscillatory contributions in the DO and ED systems wehave grouped all contributions into either oscillatory or static asshown in Figure 2a. If we consider only the two main vibrationalsub-states in DO, the 2DPE spectrum will have only ESE and GSBcontributions, while ED additionally has ESA. As a function of t2,the DO system has 8 oscillatory and 8 static configurations, whichorganize into six peaks, while the ED system has only 4 oscillatoryand 8 static contributions which give four peaks. The net result isthat the diagonal peaks in the kI and cross-peaks in the kII signalsare non-oscillatory in the ED, while all peaks except the upperdiagonal peak in kI are oscillatory in the DO. We thus find signifi-cant differences in oscillatory peaks between ED and DO systems.

An important additional parameter to consider is a phase ofoscillation. Eq. (1) can be analytically integrated for GjðtjÞ /expð�iejtj � cjtjÞ. For a single contribution Sn giving rise to a peakat ðx1;x3Þ ¼ ð�je1j; e3Þ we shift the origin of ðx1;x3Þ plot to thepeak center by introducing the displacements (x1 þ e1 ¼ �s1,x3 � e3 ¼ s3 for the rephasing pathways, while x1 � e1 ¼

.

Figure 3. The amplitudes of oscillatory peaks of 2D spectra of the DO model for kI

and kII signal. Note that the negative amplitude denotes a phase shift of p of theoscillation.

42 V. Butkus et al. / Chemical Physics Letters 545 (2012) 40–43

s1;x3 � e3 ¼ s3 for the nonrephasing). For c � c1 � c3 we get thepeak profile

Snðs3; t2; s1Þ ¼ AnLðs1; s3Þe�c2t2 cosð e2j jt2 þ /ðs1; s3ÞÞ; ð3Þ

where the lineshape and phase for the kI (upper sign) and kII (lowersign) signals are

Lðs1; s3Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½c2 � s1s3�2 þ c2ðs3 � s1Þ2

qðs2

1 þ c2Þðs23 þ c2Þ

; ð4Þ

/ðs1; s3Þ ¼ sgn e2ð Þ arctancðs3 � s1Þ�s1s3 � c2ð Þ

� �: ð5Þ

The phase / and the full profile for An ¼ 1 and t2 ¼ 0 are shown inFigure 2c. The rephasing and non-rephasing configurations areobtained by flipping the direction of the s1 axis. At the center ofthe peak (s1 ¼ s3 ¼ 0), we have / ¼ 0, leading to Sn / cosð e2j jt2Þ.However, for (s1 – 0; s3 – 0) we find Sn / cosð e2j jt2 þ /ðs1; s3ÞÞ with/ðs1; s3Þ – 0. Thus, the displacement from the peak center determinesthe phase of the spectral oscillations. Note that the sign of the phase/ is opposite for the peaks above (e2 < 0) and below (e2 > 0) thediagonal line, and this applies for all contributions.

The whole 2DPE spectrum is a sum of all relevant contributions.Assuming that all dephasings are similar, different contributions tothe same peak will have the same spectral shape and they may besummed. We can then simplify the 2DPE plot by writing the signalas a sum of peaks

P, which have static (from populations) and

oscillatory (from coherences) parts:

Sðx3; t2;x1Þ ¼ e�c2t2X

i;jLijðx1;x3Þ

� Apij þ Ac

ij cosðjxijjt2 þ /ijðx1;x3ÞÞh i

: ð6Þ

Here xij is the characteristic oscillatory frequency of a peak (ij),Ap

ijðt2Þ and Acijðt2Þ are the real parts of orientationally-averaged pre-

factors of population and coherence (electronic or vibronic) contri-butions, respectively. The spectral lineshape is given by Lijðx1;x3Þ.Here we clearly identify the oscillatory amplitude and its phase fora specific peak.

To apply this expression to our systems, we assume a typicalsituation where the spectrum of the laser pulses is tuned to thecenter of the absorption spectrum and the limited bandwidth se-lects the two strongest absorption peaks. In the 2DPE spectratwo diagonal and two off-diagonal peaks for ED and DO are ob-served. Indices i and j in Eq. (6) run over the positions of the peaksand thus can be (1,1), (1,2), (2,1), and (2,2). For clarity we study thespectral dynamics with t2 at the short delays, t2 c�1

2 , and usenotations A; L for the kI signal and eA; eL for the kII signal.

The transition dipole properties of the ED results in the picturewhere all static amplitudes of the ED are positive and Ap

11 ¼ eAp11,

Ap22 ¼ eAp

22;Ap12 ¼ Ap

21 ¼ eAp21 ¼ eAp

12. The oscillatory amplitudes areequal: Ac

12 ¼ Ac21 ¼ eAc

11 ¼ eAc22. Such relationships are obtained by

considering the all-parallel organization of polarization of incom-ing electric fields and neglecting the bi-exciton binding energy.The spectral beats with t2 can thus only have the same phases inthe kI or kII spectrum, when measured at peak centers. Addition-ally the oscillatory ESE and ESA parts in ED cancel each other ifD ¼ 0 and their broadenings are equal. As these relationships donot depend on coupling J and transition dipole orientations, allED systems should behave similarly. By studying the whole param-eter space, it can also be shown that these relations hold for a het-ero-dimer.

The amplitude-relationships, however, are different for the DOsystem. The amplitudes AðcÞij of the oscillatory peaks are plotted inFigure 3 as a function of the HR factor, where now we include allvibrational levels in the jgmi and jeni ladders. For kI , the amplitudesAc

11 and Ac22 maintain the opposite sign when HR < 2 and are both

positive when 2 < HR < 3 (note that Ac22 ¼ 0 when only two vibra-

tional levels are considered in Figure 2b). The oscillation ampli-tudes Ac

11 and Ac22 change sign at HR ¼ 1. Amplitudes Ac

12 and Ac21

are always positive. Spectrum oscillations with t2 for both diagonalpeaks in the kII signal will be in-phase for the whole range of theHR factor. The same pattern holds for the 1-2 and 2-1 cross-peaks,which will oscillate in-phase, but will be of opposite phase com-pared to the diagonal peaks in the region of HR < 1. Note thatthe sign of amplitudes changes with the HR factor, since the over-lap integral between vibrational wavefunctions can be both posi-tive and negative. The amplitudes of static contributions arepositive in the whole range of parameters and are identical for bothkI and kII signals.

We thus find very different behavior of oscillatory peaks of DOand ED systems. The above analysis applies for the central posi-tions of the peaks, which may be difficult to determine if thebroadening is large. Note that the phase / varies from �p=2 toþp=2 (Eq. (3) and Figure 2c) when probing in the vicinity of thepeak. However, / ¼ 0 along the diagonal line for kI and along theanti-diagonal line for kII. These lines can thus be used as guidelinesfor reading phase relations of distinct peaks in the 2DPE spectrum.For instance, the two diagonal peaks can be calibrated by readingtheir amplitudes at the diagonal line, or the two opposite cross-peaks can be compared by drawing anti-diagonal lines.

The 2DPE spectra for both DO and ED systems calculated byincluding phenomenological relaxation and Gaussian laser pulseshapes [14,15] are plotted in Figure 4. The structure and the t2 evo-lution of the spectra illustrate the dynamics discussed above andclearly shows the distinctive spectral properties of the vibronicvs. electronic system: (i) diagonal peaks in the kI signal are oscillat-ing in DO, but only exponentially decaying in ED (the oscillatorytraces come from the overlapping tails of off-diagonal peaks), (ii)the relative amplitude of oscillations is much stronger in DO ascompared to ED, where the ESA and ESE cancellation suppressesthe oscillations, (iii) opposite oscillation phases are observed inDO, while all peaks oscillate in-phase in ED; such behavior of theDO model arises from n > 0 vibronic states which are not presentin the ED system.

The up-to-date experiments are capable of creating broad-bandpulses [9]. Thus, the overtones in DO can be excited and beats ofnx0 frequencies (n is integer) observed. These may become impor-tant in the case of large HR factors. Such frequencies are absent inthe ED system, since only one oscillatory frequency, equal to 2J isavailable. In case of the finite pulses the pulse overlap regionmay affect the phase of the oscillatory components in the 2D spec-trum. This effect should be considered when pulse bandwidth iscomparable to the system absorption spectrum.

The analysis presented in this article provides a clear physicalpicture of electronic and vibronic coherence beatings in 2DPE spec-tra. We are able to discriminate weakly damped electronic and vib-

Figure 4. 2DPE spectra and peak values of the DO and ED as the function ofpopulation time t2, of the kI and kII signals. Spectra are normalized to the maxima ofthe total spectra of the DO and ED. With the chosen set of parameters the oscillationfrequencies are x0 ¼ 2J ¼ 600 cm�1.

V. Butkus et al. / Chemical Physics Letters 545 (2012) 40–43 43

ronic coherent wavepackets in molecular systems based on funda-mental theoretical considerations. Dynamics of diagonal peaks andcross-peaks as well as relative phase between them in the rephas-ing signal can now be classified for vibrational and excitonic sys-tems as follows. (i) Static diagonal peaks and oscillatory off-diagonal peaks signify pure electronic coherences, not involved inenergy transport. (ii) Oscillatory diagonal peaks in accord withoff-diagonal peaks (0 or p phase relationships) signify vibronic ori-gins. The oscillation phase is 0 for electronic coherences and 0 or pfor vibronic coherences. These outcomes hold if the signal isprobed at the very centers of the spectral resonances. The observed

phase of the beatings varies as the signal is recorded away from thecenter of an oscillating peak.

Our results might be useful in analysis of recently observedbeatings in molecular systems. For instance phase relations ofthe beatings detected at separate points in the vicinity of the samecross-peak of the photosynthetic LH2 complex [16] might be theresult of measurement away from the peak center (see Figure2b). The issue of probing away from the peak centers also appliesto the opposite-phase beatings reported by Collini et al. [7]. Ouranalysis thus shows that the detailed phase relationships in thetwo dimensional spectra may be of critical importance. From thisstudy it also follows that the oscillations induced by the vibrationalmodes are expected to be much stronger than the ones from elec-tronic coherences when the bi-exciton binding energies are small.This is usually the case for the photosynthetic aggregates [17]. Byhelping to identify spectral beats in such systems, the presentedanalysis should facilitate answering the question of importanceof electronic coherences in excitonic energy transfer, its efficiencyand robustness.

Acknowledgement

This research was funded by the European Social Fund underthe Global Grant measure.

References

[1] P.F. Tian, D. Keusters, Y. Suzaki, W.S. Warren, Science 300 (2003) 1553.[2] X. Li, T. Zhang, C.N. Borca, S.T. Cundiff, Phys. Rev. Lett. 96 (2006) 057406.[3] G.S. Engel, T.R. Calhoun, E.L. Read, T.K. Ahn, T. Mancal, Y.C. Cheng, R.E.

Blankenship, G.R. Fleming, Nature 446 (2007) 782.[4] E. Collini, G.D. Scholes, Science 323 (2009) 369.[5] F. Caruso, A.W. Chin, A. Datta, S.F. Huelga, M. Plenio, J. Chem. Phys. 131 (2009)

105106.[6] P. Rebentrost, M. Mohseni, A.A. Guzik, J. Phys. Chem. B 113 (2009) 9942.[7] E. Collini, C.Y. Wong, K.E. Wilk, P.M.G. Curmi, P. Brumer, G.D. Scholes, Nature

463 (2010) 644.[8] A. Nemeth, F. Milota, T. Mancal, V. Lukeš, J. Hauer, H.F. Kauffmann, J. Sperling, J.

Chem. Phys. 132 (2010) 184514.[9] N. Christensson, F. Milota, J. Hauer, J. Sperling, O. Bixner, A. Nemeth, H.F.

Kauffmann, J. Phys. Chem. B 115 (2011) 5383.[10] D.B. Turner, K.E. Wilk, P.M.G. Curmi, G.D. Scholes, J. Phys. Chem. Lett. 2 (2011)

1904.[11] V. May, O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems,

Wiley-VCH, 2011.[12] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford University

Press, New York, 1995.[13] H. van Amerongen, L. Valkunas, R. van Grondelle, Photosynthetic Excitons,

World Scientific, Singapore, 2000.[14] D. Abramavicius, L. Valkunas, S. Mukamel, Europhys. Lett. 80 (2007) 17005.[15] D. Abramavicius, V. Butkus, J. Bujokas, L. Valkunas, Chem. Phys. 372 (2010) 22.[16] E. Harel, G.S. Engel, Proc. Natl. Acad. Sci. USA 109 (2012) 706.[17] D. Abramavicius, B. Palmieri, S. Mukamel, Chem. Phys. 357 (2009) 79.