time-domain chirally-sensitive three-pulse coherent probes ...the nee provide a uniﬁed treatment...

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Chemical Physics 318 (2005) 50–70

Time-domain chirally-sensitive three-pulse coherent probesof vibrational excitons in proteins

Darius Abramavicius *, Shaul Mukamel

Chemistry department, University of California, Irvine, CA 92697-2025, United States

Received 24 March 2005; accepted 17 June 2005Available online 29 September 2005

Abstract

The third-order optical response of Bosonic excitons is calculated using the Green�s function solution of the nonlinear excitonequations (NEE), which establish a quasi particle-scattering mechanism for optical nonlinearities. Both time-ordered andnon-ordered forms of the response function which represent time and frequency domain techniques, respectively, are derived.New components of the response tensor are predicted for isotropic ensembles of periodic chiral structures to first order in the opticalwavevector. The nonlocal nonlinear response function is calculated in momentum space. The finite exciton–exciton interactionlength is used to greatly reduce the computational effort. Applications arc made to coupled anharmonic vibrations in the amideI infrared band of peptides. Chirally sensitive and non-sensitive signals for a helices and antiparallel b sheets are compared.� 2005 Elsevier B.V. All rights reserved.

Keywords: Nonlinear response; Four-wave mixing; Excitons; Nonlinear exciton equations

1. Introduction

Calculating the nonlinear optical response of large molecules using conventional sum-over-states expressions is achallenging task, since the number of states accessible by multiple quantum excitations increases rapidly with systemsize [1,2]. For example, the Frenkel exciton model for M coupled three-level systems has M one-exciton states andM� ðMþ 1Þ=2 two-exciton states. Diagonalizing the two-exciton Hamiltonian is the bottleneck in numerical simula-tions. The nonlinear exciton equations (NEE) [3–5] offer an alternative exact method for computing the nonlinear opti-cal response of systems whose Hamiltonian conserve the number of excitons. The NEE establish an exciton scatteringmechanism and provide a practical algorithm for computing the third-order optical response, totally avoiding the cal-culation of two-exciton states. The NEE were first developed by Spano and Mukamel [6–8] and applied for four-wavemixing of coupled two-level [3,9,10] and three-level [1,5,11,12] molecules. The local-field-approximation was general-ized by adding two-exciton variables which properly account for double quantum resonances. Additional dynamicalvariables have subsequently been added, allowing to describe population transport via the Redfield equation for thereduced exciton density matrix [13,14]. The NEE were further extended to particles with arbitrary commutation rela-tions, and to Wannier excitons in semiconductors [4,15,16].

0301-0104/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2005.06.046

* Corresponding author.E-mail addresses: [email protected] (D. Abramavicius), [email protected] (S. Mukamel).

mailto:[email protected]

mailto:[email protected]

D. Abramavicius, S. Mukamel / Chemical Physics 318 (2005) 50–70 51

We have recently studied the frequency-domain third-order susceptibility of isotropic chiral exciton systems and cal-culated the leading terms of its tensor elements going beyond the dipole approximation [17]. In this paper, we applythese results to the time-domain response function in a collinear field configuration.

The NEE provide a unified treatment of vibrational and electronic excitons. The Frenkel excitons corresponding toelectronic excitations are Paulions whose non-Boson operator statistics results in nonlinear hard-core repulsive inter-actions between excitons. The NEE were originally derived using a Bosonization procedure [6–8,13,18–20], wherebythe Pauli exciton operators are replaced by Boson operators and a repulsive potential is added to the Hamiltonian.This results in a soft-core Boson model with a finite anharmonicity D; the original hard-core Boson Hamiltonian isrecovered by setting D!1.

This article focuses on vibrational excitons, which are intrinsically Bosons [11,12,17] and the finite anharmonicitiesserve as the source of nonlinearities. Third-order impulsive optical techniques performed with linearly polarized light(Fig. 1) are commonly used to probe ultrafast processes in isotropic systems. The corresponding nonlinear responsefunction Sð3Þm4m3m2m1

is a fourth-rank tensor [21] which relates the third-order polarization, P(3)(r, t), at position r and timet to the optical field E(r 0, t 0); mi = x,y,z are the polarization components of the ith laser pulse in the lab frame. In gen-eral, the response function has 34 tensor elements. For isotropic systems in the dipole approximation, only those ele-ments with dm4m3dm2m1 ; dm4m2dm3m1 ; and dm4m1dm3m2 , (e.g., xxxx and zzyy) survive rotational averaging, and three linearlyindependent elements xxyy, xyxy and xyyx, are necessary to describe the response for an arbitrary pulse polarizationconfiguration [22,23]. However, other elements with an odd number of repeating indices (such as xxxy, zzxy), whichvanish in this approximation due to isotropic symmetry, become finite when the dipole approximation is relaxed. Wehave recently shown that these elements are chirally sensitive, i.e., appear only in chiral molecules (‘‘handed’’ systemswhich are distinct from their mirror images [23,24]), and vanish in racemates, equal mixtures of molecules with oppo-site sense of chirality.

The dipole approximation implies that the optical electric field is uniform across the molecule, thus, its phase factorexp(ikr), where k is the wavevector, does not affect the response and can be ignored. However, the wavevector doesplay an important role in the spectroscopy of chiral molecules, where the variation of the phase induces new tensorcomponents of the response function to first order in the wavevector, which vanish in the dipole approximation.The various tensor elements can be probed directly using different time-domain techniques which control the sequencesof optical interactions.

The difference in absorption of left- and right-handed circularly polarized light [23,25–27] known as circular dichro-ism (CD) is the simplest example of a wavevector-induced signal and is related to the Sð1Þxy elements of the linear re-sponse tensor (when the field propagates along z). The technique is sensitive to molecular structure and is

Fig. 1. (Top) Time-domain third-order experiments: three short laser pulses with wavevectors k1, k2 and k3 generate a nonlinear polarization withwavevector kS. (Bottom left) Structures of the peptide backbone in a helix and antiparallel b sheet structures: green – C atoms, blue – N atoms, red –O atoms. C@O responsible for amide I mode (1600–1700 cm�1) are emphasized. (Bottom right) Energy level scheme of excitonic system with oneground state, a manifold of one-exciton states (e) and a manifold of two-exciton states (f). (For interpretation of the references in color in this figurelegend, the reader is referred to the web version of this article.)

52 D. Abramavicius, S. Mukamel / Chemical Physics 318 (2005) 50–70

extensively used for protein structure determination both in the UV (180–220 nm, n–p* and p–p* transitions) and theIR (1000–3500 cm�1 which covers most of the amide vibrational bands) [28–32]. Pattern-recognition and decomposi-tion algorithms are used to distinguish between a-helical and b-sheet formations using electronic [33–35] and vibra-tional CD [36].

Polypeptides often have almost translationally invariant secondary structures (helices, sheets and strands)which form periodic arrays of localized vibrations. Analyzing secondary structural motiffs, thus, provides specificinformation which can be used for studying globular proteins with different secondary structures. Periodicity canbe used to greatly reduce the computational effort, as is the case for electronic excitations in molecular crystalsand semiconductor superlattices [16,37–42]. Due to different translational properties of one-exciton and two-exciton states in the Frenkel exciton model, multi-exciton resonances cannot be generally calculated analyticallyeven for infinite periodic systems. The NEE only require the one-exciton states, and yield closed expressions forinfinite periodic systems, where translational symmetry helps reduce the problem size even further. Relaxing thedipole approximation using the NEE, is straightforward. The CD spectra of molecular aggregates modeled ascollections of coupled electric dipoles were calculated to first order in wavevectors [43]. This model has beenapplied to biological light harvesting antennae and cylindrical aggregates [44,45]. By extending this procedureto the nonlinear response, we obtain the complete set of tensor elements for the response function of infiniteperiodic systems.

In Section 2, we present the time-domain expressions for the third-order optical response. The Hamiltonian and theNEE for vibrational excitons are given in Section 3 and the third-order response function is derived in Section 4. Twotechniques for probing one-exciton and two-exciton resonances are discussed in Section 5. The two-dimensional infra-red frequency correlation signals of two typical structural motifs of polypeptides: the one-dimensional a helix and thetwo-dimensional antiparallel b sheet (Fig. 1) in the amide I region (1550–1750 cm�1) are compared. The results arediscussed in Section 6. The linear absorption is calculated in Appendix A. Rotational averages for isotropic systemsare given in Appendix B. The nonlinear signals are calculated in Appendices C (four-wave mixing in various phasematching directions) and D (pump–probe). The frequency-domain signal is given in Appendix E and the exciton scat-tering matrix for an infinite periodic system is presented in Appendix F.

2. Time-domain optical response of excitons

The optical response of molecules is determined by the induced polarization, which serves as the source in the Max-well equations for the generated signal field. The response functions SðnÞmnþ1mn��m1 are system property-tensors, which allowto calculate the induced polarization for an arbitrary incoming pulse configuration [2]:

PðnÞmnþ1ðxnþ1Þ ¼

Xmn;...;m1

Zdxn � � �

Zdx1 S

ðnÞmnþ1mn��m1ðxnþ1; xn; . . . ; x1ÞEmnðxnÞ � � �Em1ðx1Þ. ð1Þ

Here E is the optical electric field vector and m = x,y,z denote its Cartesian components in the lab frame; x = (r, t) is thespace–time vector and

Rdx �

Rdr

Rdt, where the r integration runs over the molecular volume, while the t integra-

tion is from �1 to +1. x represent the times and coordinates of the interactions with the optical pulses.By causality, S(n) vanishes unless t1 � � � tn precede tn+1 for all n. We focus on time-domain experiments, where the

interaction sequence is controlled by short optical pulses. We define the time-ordered response function to be finitefor tn+1 > tn > � � � > t2 > t1 and to vanish otherwise as shown in Fig. 2(a). Alternatively we can require that the responsefunction to be symmetric with respect to permutation of mjxj, j = 1,2,3. This non-time-ordered response function,which is useful for frequency-domain techniques is given in Appendix E.

The linear polarization is given by:

Pð1Þm2ðx2Þ ¼

Xm1

Zdx1 S

ð1Þm2m1ðx2; x1ÞEm1ðx1Þ; ð2Þ

where Sð1Þm2m1ðx2; x1Þ is the nonlocal linear response function. S(1) is responsible for linear absorption (see Appendix A)

and CD. Four-wave mixing (4WM) signals are described by the third-order polarization

Pð3Þm4ðx4Þ ¼

Xm3m2m1

Zdx3

Zdx2

Zdx1 S

ð3Þm4m3m2m1

ðx4; x3; x2; x1ÞEm3ðx3ÞEm2ðx2ÞEm1ðx1Þ. ð3Þ

t2 t3t1 t4

k3k2k1 k4

t

T 1= t21 T 2= t32 T 3= t 43

τ , τ , ,

τ s, , τ s

,

t2

t 3

t1

τ 1

2

τ 3

t2

t3

t1

τ 3

τ 2

τ 1

t2

t3

t1

τ 2

τ 3

τ 1

t21

t32

t 43

τ s,

τ s, ,

− k1

k2

k3

τ s,

τ s, ,

k1

k3

− k2

τ s,

τ s, ,

k1

k2

− k3

k4 k4

k4

− k1+ k2+ k3 k1− k2+ k3k1+ k2− k3

n1

n2

n3

n4

n1,n4

,

n3,

n2,

n1

n2

n3

n3,

n2,n4

,

n1,

n4

n1

n2

n3

n3,

n2,

n1,

n4,

n4

a

b

c

Fig. 2. (a) Time variables used in calculating the excitonic response. Red peaks indicate laser pulses, blue area corresponds to the exciton–excitonscattering process. t1, t2 and t3 are the first, second and third interactions with laser pulses. t4 is the signal generation time. Ti � ti+1 � ti are the delaytimes (always positive) between two interactions. s00s and s0s are the delay times of the exciton scattering. (b) Transformation from the non-orderedtime variables s1, s2 and s3 to the ordered times t1, t2 and t3 in Eqs. (24)–(26), which defines three distinct different scattering pathways. (c) Thescattering pathways: kI = �k1 + k2 + k3 involves scattering of the excitons created by k2 and k3, kII = k1 � k2 + k3 involves scattering of the excitonscreated by k1 and k3, and kIII = k1 + k2 � k3 involves scattering of the excitons created by k1 and k2. (For interpretation of the references in color inthis figure legend, the reader is referred to the web version of this article.)


We assume three short well-separated incoming pulses:

EmðxÞ ¼1

2

X3

s¼1

~EðsÞm ðt � tsÞ expðiksr� ixsðt � tsÞ þ i/sÞ þ c.c.; ð4Þ

ð2Þ
where ~Em ðt � tsÞ is the (real) envelope of pulse s centered at ts, with wavevector ks, carrier frequency xs and phase /s.When the pulses are much shorter than the relevant molecular timescale, ~Emðt � tsÞ in Eq. (2) can be approximated asEðsÞm dðt � tsÞ. Since the pulses are longer than the optical periods we must then invoke the rotating wave approximation– RWA and only retain in S(3) terms resonant with optical fields which fall within the pulse bandwidth ~xs [46]. In thisapproximation, the field amplitude in the frequency domain is taken to have a rectangular shape centered at xs withwidth ~xs. We next assume that the pulses are temporally well separated and ordered, i.e., the first pulse ~E
ð1Þinteracts at

t1, followed by ~Eð2Þ

at t2 and the last pulse is ~Eð3Þ

at t3. The third-order polarization is then given by

Pð3Þm4ð�x4Þ ¼ expð�i/3 � i/2 � i/1Þ

1

23

Xm3m2m1

Sð3Þm4m3m2m1ðk4t4;�k3t3;�k2t2;�k1t1ÞEð3Þm3

Eð2Þm2Eð1Þm1

; ð5Þ

where �x ¼ ðk; tÞ; tj and kj now coincide with the central pulse times and wavevectors and we have applied the space–time Fourier transform F ðk;xÞ ¼

Rdt

Rdr expðixti þ ikrÞF ðr; tÞ.


When the system is initially at equilibrium, the response function is time translationally invariant and only dependson the positive time intervals T1 = t2 � t1, T2 = t3 � t2, T3 = t4 � t3 (see Fig. 2). Space translational invariance for iso-tropic systems implies k4 = �k3 � k2 � k1. We thus denote Pð3Þm4

ð�k4; t4Þ � PkSm4ðT 3; T 2; T 1Þ with kS � �k4. There are

four independent possible signal wavevectors kS: kI = �k1 + k2 + k3, kII = +k1 � k2 + k3, kIII = +k1 + k2 � k3 andkIV = +k1 + k2 + k3.

3. The NEE for vibrational excitons

The amide vibrations of polypeptides can be modeled as N coupled anharmonic vibrational modes localized at thepeptide bonds and described by the exciton Hamiltonian:

H ¼Xm

emBymBm þ

Xm 6¼nmn

Jm;nBymBn þ

Xmn;m0n0

Umn;m0n0 BymBynBm0 Bn0 �

Zdr PðrÞ � Eðr; sÞ. ð6Þ

The creation, Bym, and annihilation, Bm, operators for mode m satisfy the Boson ½Bm; B

yn� ¼ dmn commutation relations.

The first two terms represent the free-Boson harmonic Hamiltonian, where em is the frequency of mode m, and thequadratic intermode coupling, Jm,n, is calculated in the Heitler–London approximation where off resonant B

ymByn

and BmBn terms are neglected. The third term represents a quartic anharmonicity. We assume a pairwise anharmonicinteraction, Umn;m0n0 ¼ Dm;n

4ðdmm0dnn0 þ dmn0dnm0 Þ, where Dmm is the on-site anharmonicity of the overtone band and Dnm is

the intermode anharmonicity of the combination band. These anharmonicities constitute the exciton–exciton scatter-ing potential. For D = 0, the Hamiltonian describes free Bosons, which is a linear system whose nonlinear responsevanishes identically [1]. The fourth term in the Hamiltonian represents the interaction with the optical field E(r, t),where

PðrÞ ¼Xm

dðr� rmÞlmðBym þ BmÞ ð7Þ

is the polarization operator and lm is the transition dipole moment of mode m located at rm; a vector with componentsðlx

m; lym; l

zmÞ.

The expectation value of the polarization operator which describes the vibrational response to the optical field willbe calculated by solving the NEE [3,4,7]. This hierarchy of equations of motion for exciton variables may be exactlytruncated order-by-order in the field since the molecular Hamiltonian conserves the number of excitons, and the opti-cal field creates or annihilates one exciton at a time. When pure-dephasing is neglected, the only relevant variables forthe third-order response are Bm ¼ hBmi (one-exciton) and Y mn ¼ hBmBni (two-exciton) and the NEE read [7,8]:

� ioBm

osþXn

hm;nBn ¼ ~lmðsÞ �Xl0m0n0

V ml0m0n0B�l0Y m0n0 ; ð8Þ

� ioY mn

osþXm0n0ðhðY Þmn;m0n0 þ V mn;m0n0 ÞY m0n0 ¼ ~lmðsÞBn þ ~lnðsÞBm. ð9Þ

Here hm,n = dm,nem + Jm,n(1 � dm,n) is an effective one-exciton Hamiltonian, hðY Þmn;m0n0 ¼ dm0 ;mhn;n0 þ dn;n0hm;m0 is a free-two-exciton Hamiltonian, V mn;m0n0 � Umn;m0n0 þ Unm;m0n0 ¼ Dm;n

2ðdmm0dnn0 þ dmn0dnm0 Þ is the anharmonicity matrix and

~lmðsÞ ¼ lm � Eðrm; sÞ. The polarization is given by the expectation value of Eq. (7):

Pðr; sÞ ¼Xm

dðr� rmÞlmBmðsÞ þ c.c. ð10Þ

The nonlinearities in these equations originate from the anharmonicity: as indicated earlier, for V = 0 Eq. (8) is linear,and the nonlinear response vanishes (the two-exciton variable can be exactly factorized as Ymn = BmBn in Eq. (9)).

The evolution of a single-exciton created by an impulsive excitation, Bð1Þm ðsÞ, is described by the one-exciton Green�sfunction G(s):

Bð1Þm ðsÞ ¼Xm0

Gm;m0 ðsÞBð1Þm0 ð0Þ; ð11Þ

which satisfies the equation

dGm;nðsÞds

þ iXn0

hm;n0Gn0;nðsÞ ¼ dðsÞ. ð12Þ


This equation can be solved using the one-exciton eigenenergies, En, and eigenvectors, wnm
Xn
hm;nwnn ¼ Enwnm. ð13Þ

We then get

Gm;nðsÞ ¼Xn

wnmInðsÞw�nn; ð14Þ

where

InðsÞ ¼ hðsÞ expð�iEns� cnsÞ; ð15Þ

cn is a dephasing rate of the n exciton state, and the Heavyside step function (h(s) = 0 for s < 0 and h(s) = 1 for s P 0)guarantees causality.

Similarly, we define the two-exciton Green�s function GY :

Y mnðsÞ ¼Xm0n0

GYmn;m0n0 ðsÞY m0n0 ð0Þ; ð16Þ

whose equation of motion:

dGYmn;m0n0

dtþ i

Xm00n00ðhðY Þmn;m00n00 þ V mn;m00n00 ÞGY

m00n00;m0n0 ¼ dðsÞ. ð17Þ

GY is connected to the zero-order noninteracting (V = 0) two-exciton Green�s function G by the Bethe–Salpeter equa-tion [3,4]:

GY ðsÞ ¼ GðsÞ þZ s

0

ds0Z s0

0

ds1 Gðs� s0ÞCðs0 � s1ÞGðs1Þ; ð18Þ

which defines the two-exciton scattering matrix C. Both the two-exciton Green�s function and the scattering matrix aretetradic matrices; like G(s) the scattering matrix is causal as well. G can be factorized into a product of one-excitonGreen�s functions, Gmn;m0n0 ðsÞ ¼ Gm;m0 ðsÞGn;n0 ðsÞ.

The frequency-domain scattering matrix C(x) obtained by the Fourier transform, CðxÞ ¼Rdt expðixtÞCðtÞ, is cal-

culated in Appendix F [3,17]:

CðxÞ ¼ �iV ð1þ iGðxÞV Þ�1; ð19Þ
where
Gmn;m0n0 ðxÞ ¼Xnn0

wnmwn0nInn0 ðxÞw�nm0w�n0n0 ; ð20Þ

with

Inn0 ðxÞ ¼i

x� En � En0 þ iðcn þ cn0 Þ. ð21Þ

Calculating the scattering matrix requires the inversion of the matrix D ¼ 1þ iGðxÞV whose matrix elements:

Dmn;ijðxÞ ¼ dmidnj þ iXm0n0

Gmn;m0n0 ðxÞV m0n0;ij. ð22Þ

The required numerical effort can be reduced considerably for periodic systems and when the short range nature ofanharmonicities is taken into account [3].

4. The nonlinear optical response: Green�s function solution of the NEE

The nonlinear response is calculated by an order-by-order expansion of the NEE variables in the field using the exci-ton Green�s functions, where the optical field and the lower order variables serve as the sources. SettingBm ¼ Bð1Þm þ Bð2Þm þ Bð3Þm þ � � � and Y mn ¼ Y ð1Þmn þ Y ð2Þmn þ Y ð3Þmn þ � � � we get to third order [4,17]:


Bð3Þn4ðs4Þ ¼ 2i

Z 1

�1ds00

Z 1

�1ds0

Z 1

�1ds3

Z 1

�1ds2

Z 1

�1ds1

Xn1n2n3

Xn01n02n03n04

hðs2 � s1ÞCn04n03;n0

2n01ðs00 � s0Þ

� Gn4;n04ðs4 � s00ÞGyn0

3;n3ðs00 � s3ÞGn0

2;n2ðs0 � s2ÞGn0

1;n1ðs0 � s1Þ~ln3ðs3Þ~ln2ðs2Þ~ln1ðs1Þ. ð23Þ

The s 0 and s00 variables denote the times of the first and the last exciton–exciton interaction, respectively, as shown inFig. 2(a). The third-order polarization is finally obtained from Eqs. (10) and (23):

P ð3Þm4ðr4; s4Þ ¼ 2i

Zdr3

Zdr2

Zdr1

Z 1

�1ds3

Z 1

�1ds2

Z 1

�1ds1

Xn4n3n2n1

hMm4m3m2m1n4n3n2n1

ðr4; r3; r2; r1Þi

�Z 1

�1ds00

Z 1

�1ds0

Xn01n02n03n04

hðs2 � s1ÞCn04n03;n0

2n01ðs00 � s0ÞGn4;n04

ðs4 � s00ÞGyn03;n3ðs00 � s3ÞGn0

2;n2ðs0 � s2Þ

� Gn01;n1ðs0 � s1ÞEm3ðr3; s3ÞEm2ðr2; s2ÞEm1ðr1; s1Þ þ c.c.; ð24Þ

where

Mm4m3m2m1n4n3n2n1

ðr4; r3; r2; r1Þ ¼ dðr4 � rn4Þdðr3 � rn3Þdðr2 � rn2Þdðr1 � rn1Þlm4n4lm3n3lm2n2lm1n1; ð25Þ

‘‘c.c.’’ is the complex conjugate and h� � �i denotes rotational averaging (Appendix B). It is important to note that unliket1, t2 and t3, the integration variables s1, s2 and s3 do not have any particular time ordering.

Since the Green�s functions are retarded (i.e., they vanish for negative time arguments) only three sequences of inter-action times contribute to Eq. (24): (i) s2 > s1 > s3, (ii) s2 > s3 > s1 and (iii) s3 > s2 > s1. In each case, we switch to adifferent set of time-ordered variables: (i) t4 = s4, t3 = s2, t2 = s1, t1 = s3, (ii) t4 = s4, t3 = s2, t2 = s3, t1 = s1 and (iii)t4 = s4, t3 = s3, t2 = s2, t1 = s1 (Fig. 2(b)). Eq. (3) together with Eq. (24) then result in the following three contributionsto the response function:

Sð3Þm4��m1ðx4; . . . ; x1Þ ¼ SðIÞm4��m1ðx4; . . . ; x1Þ þ SðIIÞm4��m1ðx4; . . . ;x1Þ þ SðIIIÞm4��m1ðx4; . . . ; x1Þ þ c.c. ð26Þ

Each term now corresponds to one particular interaction sequence (S(I) is obtained from (i), S(II) – from (ii) and S(III) –from (iii)). These are given by:

SðIÞm4��m1ðx4; . . . ; x1Þ ¼ 2iXn4��n1

hMm4m3m2m1n4n3n2n1

ðr4; r3; r2; r1ÞZ t43

0

ds00s

Z t00s

0

ds0sX

n04n03n02n01

Cn04n01;n0

3n02ðs00s � s0sÞ

� Gn4;n04ðs0sÞGn0

3;n3ðt43 � s00s ÞGn0

2;n2ðt42 � s00s ÞG

yn01;n1ðt41 � s0sÞ; ð27Þ

SðIIÞm4��m1ðx4; . . . ; x1Þ ¼ 2iXn4��n1

hMm4m3m2m1n4n3n2n1

ðr4; r3; r2; r1ÞiZ t43

0

ds00s

Z t00s

0

ds0sX

n04n03n02n01

Cn04n02;n0


� Gn4;n04ðs0sÞGn0

3;n3ðt43 � s00s ÞG

yn02;n2ðt42 � s0sÞGn0

1;n1ðt41 � s00s Þ; ð28Þ

SðIIIÞm4��m1ðx4; . . . ; x1Þ ¼ 2iXn4��n1

hMm4m3m2m1n4n3n2n1

ðr4; r3; r2; r1ÞiZ t42

0

ds00s

Z t00s

0

ds0sX

n04n03n02n01

Cn04n03;n0


� Gn4;n04ðs0sÞG

yn03;n3ðt43 � s0sÞGn0

2;n2ðt42 � s00s ÞGn0

1;n1ðt41 � s00s Þ. ð29Þ

Here s0s denotes the time interval between the polarization detection and first exciton scattering event, while s00s denotesthe interval between the detection and the last exciton scattering event, as shown in Fig. 2(a). Using these time-orderedexpressions, we can switch to new variables representing the time intervals between interactions tij = ti � tj with i > j:t43 = T3, t42 = T3 + T2, t41 = T3 + T2 + T1. Thus, all terms in the response function only depend on these three positivetime intervals. So far, the three terms in the response function merely represent different time orderings in the integra-tions, however, we will shortly see that they represent distinct optical signals.

The time intervals in Eqs. (27)–(29) and their relations to the actual interaction times are depicted in Fig. 2. Theresponse can be interpreted using Fig. 2(c): Let us consider S(III): three interactions with the optical fields at timest1 < t2 < t3 generate three quasi-particles (two with positive oscillation frequency and one – with negative) whose evo-lution is described by the one-exciton Green�s functions Gn0

1;n1 , Gn0

2;n2 and Gyn0

3;n3. The two positive frequency quasi-

particles at and n01 and n02 are scattered by Cn04n03;n0

2n01to n03 and n04. One of the two excitons, n03, corresponds to the exciton

generated by the third field, n03 n3. The other exciton, n04 ! n4, generates the optical response. S(I) and S(II) differ by

the scattering sequence of events as shown in Fig. 2(c) and may be interpreted similarly.


Using Eq. (3) and invoking the same approximations used in Section 2, we can relate the wavevectors and times inthe response functions (Eqs. (27)–(29)) to the wavevectors and times of the optical pulses. Assuming that the carrierfrequencies of all pulses are resonant with the one-exciton manifold, we select the resonances in the response functionsusing the RWA and express the nonlinear polarization for the signal wavevectors kI, kII and kIII. To find the wave-vector dependence, we transform Eqs. (27)–(29) to momentum space using F(k) = �drF(r)exp(ikr). Then:

PkIm4ðT 3; T 2; T 1Þ ¼

1

23expðþi/3 þ i/2 � i/1Þ

Xm3m2m1

SkIm4m3m2m1

ðT 3; T 2; T 1ÞEð3Þm3Eð2Þm2

Eð1Þm1; ð30Þ

PkIIm4ðT 3; T 2; T 1Þ ¼

1

23expðþi/3 � i/2 þ i/1Þ

Xm3m2m1

SkIIm4m3m2m1


Eð1Þm1; ð31Þ

PkIIIm4ðT 3; T 2; T 1Þ ¼

1

23expð�i/3 þ i/2 þ i/1Þ

Xm3m2m1

SkIIIm4m3m2m1


Eð1Þm1; ð32Þ

where we have denoted

SkIm4m3m2m1

ðT 3; T 2; T 1Þ � SðIÞm4m3m2m1ððk1 � k2 � k3Þt4; k3t3; k2t2;�k1t1Þ; ð33ÞS

kIIm4m3m2m1

ðT 3; T 2; T 1Þ � SðIIÞm4m3m2m1ððk2 � k1 � k3Þt4; k3t3;�k2t2; k1t1Þ; ð34Þ

SkIIIm4m3m2m1

ðT 3; T 2; T 1Þ � SðIIIÞm4m3m2m1ððk3 � k2 � k1Þt4;�k3t3; k2t2; k1t1Þ; ð35Þ

We, thus, find that the response function S(I) generates the kI signal, S(II) generates kII and S(III) generates kIII. The

fourth possible signal PkIV vanishes for the present model since it has no transition dipole connecting the three-excitonstates with the ground state. The three terms in Eqs. (27)–(29) were obtained by a simple bookkeeping of time vari-ables. The RWA has connected these terms with the impulsive signals in Eqs. (30)–(35). Each of the three signals isthus given by a single term.

Eqs. (27)–(29) are used in Appendix C to calculate the signals in the eigenstate basis. The frequency-domain scatter-ing matrix (see Appendix F) may be used to simplify the response functions in Eqs. (C.6)–(C.8). The three signals aregiven by Eqs. (C.11), (C.14) and (C.17). The sequential pump–probe spectrum is calculated in Appendix D.

By transforming all time variables to the frequency domain, SðX3;X2;X1Þ ¼R10

dT 3

R10

dT 2

R10

dT 1

SðT 3; T 2; T 1Þ expðiX3T 3 þ iX2T 2 þ iX1T 1Þ (see Appendix C) Eqs. (33)–(35) give:

SkIm4m3m2m1

ðX3;X2;X1Þ ¼ 2iXn4��n1

hdm4n4ðk1 � k2 � k3Þdm3�

n3ð�k3Þdm2�

n2ð�k2Þdm1

n1ð�k1Þi

� INn1n2ðX2ÞI�n1ð�X1ÞIn4ðX3ÞCn4n1;n3n2ðX3 þ En1 þ icn1ÞIn3n2ðX3 þ En1 þ icn1Þ; ð36ÞSkII

m4m3m2m1ðX3;X2;X1Þ ¼ 2i

Xn4��n1


n3ð�k3Þdm2

n2ð�k2Þdm1�

n1ð�k1Þi

� INn2n1ðX2ÞIn1ðX1ÞIn4ðX3ÞCn4n2;n3n1ðX3 þ En2 þ icn2ÞIn3n1ðX3 þ En2 þ icn2Þ; ð37ÞS

kIIIm4m3m2m1

ðX3;X2;X1Þ ¼ 2iXn4��n1

hdm4n4ðk3 � k2 � k1Þdm3

n3ð�k3Þdm2�

n2ð�k2Þdm1�

n1ð�k1Þi

� In1ðX1ÞIn4ðX3ÞI�n3ðX2 � X3Þ½Cn4n3;n2n1ðX2ÞIn2n1ðX2Þ� Cn4n3;n2n1ðX3 þ En3 þ icn3ÞIn2n1ðX3 þ En3 þ icn3Þ�; ð38Þ

where we have defined the exciton transition dipole for state n

dmnðkÞ ¼

Xm

eikrmlmmwnm ð39Þ

and

INn2n1ðXÞ ¼i

Xþ En2 � En1 þ iðcn2 þ cn1Þ. ð40Þ

The application of the Green�s functions expressions to periodic infinite systems is straightforward [17]: the summa-tions over one-exciton eigenstates are replaced by summations over different Davydov exciton bands at zero momen-tum. The scattering matrix of infinite systems which involves all possible momenta of different Davydov bands is givenin Appendix F.


5. Application to the amide I band of peptides

We have applied the present theory to the amide I vibrations of two ideal structural motifs of polypeptides: a helices(one dimensional) with 18 amide residues in a unit cell, and antiparallel b sheets (two dimensional) with 4 residues perunit cell. The structural and coupling parameters were reported earlier [17,47]. The anharmonicity Dmn is local(D = �16 cm�1 for m = n and zero otherwise) and the same dephasing rate, cn = 3 cm�1 was assumed for all excitons.We used 100 cells in each dimension to calculate the scattering matrix with periodic boundary conditions (Eq. (F.2)).

The linear absorption of both motifs presented in Fig. 3 shows two absorption peaks for the a helix and the b sheet.For the a helix, the longitudinal transition – along the helix axis gives a peak at 1642 cm�1, and transverse – in theplane perpendicular to the helical axis gives a weaker peak at 1661 cm�1 [47]. For the b sheet, the main peak at1632 cm�1 is horizontal (parallel to the sheet surface). The other weak peak at 1700 cm�1 consists of two transitions[47]: vertical (perpendicular to the sheet surface transition) at 1707 cm�1 and horizontal at 1699 cm�1.

Signals were calculated for a collinear field configuration where, k1, k2 and k3, propagate along z. We further as-sume all fields to have the same carrier frequency so that |kj| = xs/c for all kj (j = 1,2,3) where c is the speed of light.The absolute magnitudes of the SkI

m4m3m2m1ðX3; T 2 ¼ 0;X1Þ and SkIII

m4m3m2m1ðX3;X2; T 1 ¼ 0Þ signals were computed using Eqs.

(C.11) and (C.17). These are one-sided Fourier transforms of the time domain kI and kIII signals.The SkI

m4m3m2m1signals for the a helix are shown in the left column in Fig. 4. The xxxx component is finite in the dipole

approximation and shows one major longitudinal peak associated with the longitudinal transition [47] and a muchweaker 20 cm�1 blue shifted transverse transition. The crosspeaks between these two peaks are symmetric with respectto the diagonal and have roughly the same amplitudes. This signal resembles our previous calculation of large (90 res-idue) helices [47], which could be treated as infinite helices with small edge defects.

This xxxx pattern is changed in the chirally sensitive xxxy component. The diagonal peaks are suppressed. This canbe explained by noting that the distance between sites enters the signal amplitude: diagonal peaks have no such dis-tance factor since they originate from interactions with the same mode. The crosspeaks are induced by interactionsof different modes and depend on their distance. Thus, in the chiral signal the crosspeaks are amplified comparedto the diagonal peaks. Additionally, the crosspeak pattern is asymmetric since one of the crosspeaks (below the diag-onal) dominates. The other, above the diagonal, crosspeak is much weaker. The remaining two chiral componentsxxyx and xyxx also suppress the diagonal peaks, while show two crosspeaks.

The antiparallel b sheet signals displayed in the right column in Fig. 4 show a similar pattern, however only onehorizontal diagonal peak (as shown in [47]) is visible in xxxx with very weak crosspeaks shifted to much higher fre-quencies compared to the diagonal. Since there are four sites per unit cell, we generally expect four optical transitions(Davydov components). This could result in four diagonal peaks and six crosspeaks in both sides of the diagonal if alltransitions are correlated. However, xxxx is dominated by a single strong horizontal transition. Similar to the helix,this peak structure changes in the chiral component, xxxy, which shows much stronger crosspeaks than the diagonalpeaks. The strongest crosspeak is well separated from the strongest xxxx peak. Like the helix, the xxyx and xyxx con-figurations are both similar and show crosspeaks above and below the diagonal.

We next turn to the kIII technique. SkIIIm4m3m2m1

shows two-exciton resonances along X2 (double quantum coherence); theresonances along X3 originate both from one- to two-exciton resonances (e to f in Fig. 1) and from one-exciton to

Fig. 3. Linear absorption of the a helix (top) and the antiparallel b sheet (bottom).

Fig. 4. Absolute value of SkIm4m3m2m1

ðX3; T 2 ¼ 0;X1Þ signal of the a helix and antiparallel b sheet (Eq. (C.10)). Shown are one non-chiral, xxxx, andthree chiral, xxxy, xxyx, xyxx, components as indicated. Blue crosses mark the crosspeaks of xxxx. (For interpretation of the references in color inthis figure legend, the reader is referred to the web version of this article.)


ground state resonances (0 to e). The xxxx component for the a helix presented in Fig. 5 shows one major peak and aweaker peak originating from the same two-exciton resonance. The main peak of the chiral components xxxy, xxyx,xyxx is shifted compared to xxxx, indicating that different two-exciton resonances make the strongest contribution tothe signal. The xxxy and xxyx signals are very similar and show two strong peaks unlike xyxx which only shows onemajor peak. The xxxx component of the b sheet displayed in Fig. 5 has a similar structure to the a helix. The chiralxxxy component shows two peaks at larger X2 comparing to xxxx. xxyx, is very similar to xxxy, xyxx shows one ma-jor peak at X3 = 1700 cm�1.

6. Discussion

In the electric dipole approximation, the linear absorption of isotropic systems is related to the diagonal tensor ele-ments of the linear susceptibility [17,43]. Nonlinear signals calculated within this approximation then provide a limitedwindow onto the optical responses of isotropic ensembles through three independent chirally non-sensitive tensor com-ponents of the third-order response function: xxyy, xyyx, xyxy. The signal propagation direction is determined by

Fig. 5. Absolute value of the SkIII ðX3;X2; T 1 ¼ 0Þ, signal of the a helix and antiparallel b sheet (Eq. (C.16)). Shown are one non-chiral, xxxx, andthree chiral, xxxy, xxyx, xyxx, components, as indicated.


phase-matching. Going beyond the dipole approximation, we found three additional components for a collinear laserconfiguration: xxxy, xxyx, xyxx. Noncollinear configurations (which can also satisfy phase matching) lead to six non-zero elements (the three additional components are: (z)xyzz, (z)xzzy and (z)xzyz, where the first index (z) denotes thewavevector, and the other four are polarization components). These are chirally sensitive and show a different patternin the correlation plots, which reflects the polarization properties of optical transitions. For example, from Figs. 4 and5 we see that the strongest peaks originate from correlations between perpendicular transitions: longitudinal–trans-verse in the a helix and horizontal–vertical in the b sheet (see Fig. 1).

The differences between the chiral components can be rationalized by following the interaction and evolution se-quences: in SkI , X1 corresponds to the free evolution after the first interaction. Subsequently, two simultaneous inter-actions take place. Therefore, xxyx and xyxx reflect a similar excitation pattern: there is one interaction first, followedby two interactions with the perpendicular polarization. xxxy is qualitatively different: the first interaction is followedby two interactions with parallel polarizations. This is why, xxxy and xxyx show a different peak pattern. SkIII has twosimultaneous interactions first, followed by one interaction. Thus, xxxy and xxyx give very similar signals since in bothcases the first two interactions are with perpendicular fields. xyxx is qualitatively different since the first two interac-tions arc with parallel fields.


The signal further carries information regarding the redistribution of exciton amplitudes between states with differ-ent polarization properties. For instance, SkIII

xyxx originates from amplitude transfer between a state with x polarizationto a state with y polarization during T2. S

kIIIxxxy is qualitatively different since the evolution during T2 corresponds to x

polarized amplitude transfer to another x exciton. SkIxxxy again originates from amplitude transfer during T1 between

states with perpendicular (y! x) polarizations; xyxx does not require such exciton evolution.In the present work, we only included homogeneous line broadening caused by fast frequency fluctuations. Slow

fluctuations result in inhomogeneous broadening which should affect the 2D lineshapes [48,49]. The ideal peak pat-terns predicted for periodic structures can be used for structure decomposition and parameter determination ofreal structures. Dynamical information on exciton evolution can be obtained by varying T2 in SkIðX3; T 2;X1Þ. Pop-ulation transport may be incorporated using the theory developed in [13,21], but this goes beyond the scope of thispaper.

We next compare the exciton scattering mechanism offered by the NEE with the more conventional picture oftransitions among eigenstates, described by double-sided Feynman diagrams. The relation between the two for kI,kII and kIII is shown in Fig. 6. For kI the exciton coherence during T1 in both pictures is represented by the same

Fig. 6. Comparison of the exciton-scattering and the transition-among-eigenstates pictures of the three signals SkI ðX3; T 2X1Þ; S

kII ðX3; T 2X1Þand S

kIII ðX3;X2; T 1Þ. X1, X2 and X3 are the Fourier transform variables conjugate to T1, T2 and T3. Red dots represent the interaction of the systemwith the field, blue dots mark the exciton scattering space. (For interpretation of the references in color in this figure legend, the reader is referred tothe web version of this article.)


Green�s function. However, during T3 the Feynman diagrams show two independent coherence evolutions, onlyone of them involves the two-exciton states. The scattering representation is different: during T3 the evolution con-sists of scattering + two free evolutions. This is natural in the molecular basis set where the exciton pathways canbe followed in real space: the short range exciton scattering is then followed by free evolution as the excitons sep-arate. The role of the distance parameter is much less obvious in the eigenstate picture, kII is similar, except thatthe density matrix evolution during the first and the last intervals has the same frequency sign. In kIII, two exci-tons are created by the initial excitation. Thus the scattering can occur at any time. In the sum-over-states rep-resentation, this is given as independent evolutions of two-exciton coherences during T1. Both representationsof the response are equivalent as long as pure dephasing is neglected. The NEE and the corresponding responsebecome more complicated by pure dephasing which induces incoherent population transport and requires addi-tional dynamic variables [4,13,21].

The response function in Eq. (3) is not required to be time ordered, thus, we can define it to be symmetric with re-spect to permutation of different time arguments. This choice is useful for overlapping optical pulses and frequencydomain response, where time ordering of interactions is not enforced. Frequency-domain signals can be expressedusing the third-order susceptibility given in terms of the exciton scattering matrix (Appendix E). The susceptibilityis then directly related to non-time-ordered response function. Expressions for kI, kII and kIII can be derived usingthe RWA.

Our theory may be applied to other periodic (infinite) as well as to non-periodic (finite) systems. Periodicity andcyclic boundary conditions result in exciton band structure where only exciton states with zero momentum are active.This applies as long as the optical wavevector is small compared with the exciton momentum. Exciton–exciton inter-actions are the source of anharmonicities and cause quasi-particle scattering. This description may also be applied tosemiconductors and quantum superlattices [16,40,42,50]. When the exciton coherence size becomes comparable to theoptical wavelength the theory may be extended to include polariton effects [14,52].

Acknowledgments

This material is based upon work supported by the National Science Foundation Grant no. CHE-0132571 and theNational Institutes of Health Grant no. 1 ROl GM59230-10A2. This support is gratefully acknowledged. We thankWei Zhuang, for most useful discussions.

Appendix A. Linear absorption of excitons

The linear response function and the linear polarization were defined by Eq. (2). The linear absorption can be cal-culated using the linear susceptibility v(1) [17]. The linear response, especially for vibrations, is often measured in thetime domain using ultrashort laser pulses: Fourier transform of the time dependent-induced polarization (free induc-tion decay) gives the Fourier Transform Infrared (FTIR) spectrum.

We assume a single excitation pulse with envelope ~Eð0Þm ðt � t0Þ centered at t0 with the carrier frequency x0 and wave-

vector k0:

EmðxÞ ¼1

2dmm0 ~E

ð0Þm0ðt � t0Þ expðik0r� ix0ðt � t0ÞÞ þ c.c. ðA:1Þ

The linear response function of the excitonic system is given in terms of the one-exciton Green�s functions [17]:

Sð1Þm2m1ð�x2; �x1Þ ¼ i

Xn

hdm2n ðk2Þdm1�

n ð�k1ÞiInðt2 � t1Þ þ c.c.0 ðA:2Þ

Within the dipole approximation, rotational averaging (Eq. (A.2)) gives hdm2n ðk2Þdm1�

n ð�k1Þi 13dm2m1 jdnj2 [17]. Substitut-

ing Eqs. (A.1) and (A.2) into Eq. (2) and using assumptions for the field given in Section 2 we can integrateRdx1 which

leads to:

Pm0ðk2t2Þ ¼i

6Eð0Þm0

Xn

jdnj2½dðk2 þ k0ÞInðt2 � t1Þ � dðk2 � k0ÞI�nðt2 � t1Þ�; ðA:3Þ

where d(k) accounts for the translational invariance of an isotropic system. This expression shows that Pm0ðk2t2Þ, onlydepends on the delay T1 = t2 � t1 and can, thus, be denoted Pm0ðk2t2Þ � Pm0ðk2; T 1Þ. The first term in Eq. (A.3) in realspace is proportional to eik0r and, thus, describes forward propagation of the induced polarization.


The absorption spectrum is defined as the imaginary part of Fourier transform of the linear polarization with re-spect to T 1 : rAðxÞ ¼ Im

R10

dT 1 expðixT 1ÞPm0ðk2; T 1Þ, while the real part of this integral describes dispersive line-shapes. By eliminating the prefactor ð1=6ÞEð0Þm0

dðk2 þ k0Þ and keeping only resonant terms we obtain:

rAðxÞ ¼Xn

cnjdnj2

ðx� EnÞ2 þ c2n. ðA:4Þ

The CD spectrum must be calculated by going beyond the dipole approximation [17].

Appendix B. The response of isotropic ensembles

Optical fields, wavevectors and space coordinates are defined in the lab frame, while the transition dipoles and theirposition vectors are given in the molecular frame. The rotational averaging in Eqs. (C.3) and (C.5), h� � �i, needs to beperformed over the relative orientation of the two frames in order to calculate the response functions for isotropic (ran-domly oriented) ensembles of molecules [22,23]:

dm4n4ðk4Þdm3

n3ðk3Þdm2�

n2ð�k2Þdm1�n1

ð�k1Þi ¼X

n4n3n2n1

wn4n4wn3n3

w�n2n2w�n1n1heik4rn4þik3rn3þik2rn2þik1rn1lm4

n4lm3n3lm2n2lm1n1i. ðB:1Þ

In the phase-matching directions, the microscopic response functions only depend on the relative positions of mole-cules and are independent on the origin of the coordinate system. Since the coordinates rm vary only within one mol-ecule, for molecules smaller than the wavelength of light we have krm 1, and the exponential function in thetransition dipole of Eq. (39) can be expanded to first order. This leads to

heik4rn4þik3rn3þik2rn2þik1rn1lm4n4lm3n3lm2n2lm1n1i hlm4

n4lm3n3lm2n2lm1n1i þ i

Xj

kj4hrjn4l

m4n4lm3n3lm2n2lm1n1i þ i

Xj

kj3hrjn3l

m4n4lm3n3lm2n2lm1n1i

þ iXj

kj2hrjn2l

m4n4lm3n3lm2n2lm1n1i þ i

Xj

kj1hrjn1l

m4n4lm3n3lm2n2lm1n1i. ðB:2Þ

In the exciton basis, Eqs. (B.1) and (B.2) give

hdm4n4ðk4Þdm3

n3ðk3Þdm2�

n2ð�k2Þdm1�

n1ð�k1Þi ¼ hdm4

n4dm3n3dm2n2dm1n1i þ i

Xj

kj4h�d

j;m4n4

dm3n3dm2n2dm1n1i þ i

Xj

kj3h�d

j;m3n3

dm4n4dm2n2dm1n1i

þ iXj

kj2h�d

j;m2n2

dm3n3dm4n4dm1n1i þ i

Xj

kj1h�d

j;m1n1

dm3n3dm2n2dm4n4i; ðB:3Þ

where we have used the fact that wn,m are real and have defined the transition dipole vector for zero momentum excitonstate

dmn � dm

nðk ¼ 0Þ ¼Xm

lmmwn;m ðB:4Þ

and the tensorX
�dj;mn ¼
m

rjmlmmwn;m. ðB:5Þ

Eq. (B.3) requires fourth and fifth rank rotational averagings. The first term in this equation corresponds to thedipole approximation. The remaining terms which contain the wavevector and a coordinate, represent a first-order cor-rection to that approximation. These terms do not depend on the origin of the molecular frame providedk4 + k3 + k2 + k1 = 0 (i.e., the signal wavevector kS = �k4), which is the phase-matching condition.

Rotational averaging can be performed using the transformation between the lab and molecular frames [23]:

hamss � � � am11 i � hðems � asÞ � � � ðem1 � a1Þi ¼

Xas��a1

TðsÞms��m1;as��a1aass � � � a

a11 ; ðB:6Þ

where TðsÞms��m1;as��a1 ¼ hlmsas � � � lm1a1i is the average of transformation tensor where lma is the cosine of the angle betweenlaboratory frame axis m = x,y,z and molecular frame axis a = x,y,z. The necessary averages of ranks four and fivetransformation tensors, which are universal quantities independent of system geometry are given in Table 1 [22].

Table 1Isotropic rotational average tensors [22]a

Tensor element Value

Tð4Þm4��m1;a4��a1 1

30

dm4m3dm2m1dm4m2dm3m1dm4m1dm3m2

0B@

1CA

T4 �1 �1�1 4 �1�1 �1 4

0B@

1CA

da4a3da2a1da4a2da3a1da4a1da3a2

0B@

1CA

Tð5Þm5��m1;a4��a1

1

30

em5m4m3dm2m1em5m4m2dm3m1em5m4m1dm3m2em5m3m2dm4m1em5m3m1dm4m2em5m2m1dm4m3

0BBBBBBBB@

1CCCCCCCCA

T3 �1 �1 1 1 0

�1 3 �1 �1 0 1

�1 �1 3 0 �1 �11 �1 0 3 �1 1

1 0 �1 �1 3 �10 1 �1 1 �1 3

0BBBBBBBB@

1CCCCCCCCA

ea5a4a3da2a1ea5a4a2da3a1ea5a4a1da3a2ea5a3a2da4a1ea5a3a1da4a2ea5a2a1da4a3

0BBBBBBBB@

1CCCCCCCCA

a ea3a2a1 is the antisymmetric Levi–Civita tensor: ea3a2a1 , is equal to 1 for (a3,a2,a1) = (xyz), (yzx), (zxy), �1 for (a3,a2,a1) = (xzy), (yxz), (zyx) and 0otherwise.


Using Table 1, we obtain for the rotational averages of the transition dipoles:

hdm4n4dm3n3dm2n2dm1n1i ¼

Xa4a3a2a1

Tð4Þm4m3m2m1;a4a3a2a1da4n4da3n3da2n2da1n1; ðB:7Þ

h�dj;m4n4

dm3n3dm2n2dm1n1i ¼

Xa5��a1

Tð5Þjm4m3m2m1;a5a4a3a2a1�da5;a4n4

da3n3da2n2da1n1. ðB:8Þ

The remaining averages can be simply obtained by permutation of indices.

Appendix C. Time-domain FWM signals

The response functions (Eqs. (27)–(29)) can be readily calculated using the single exciton basis (Eq. (13)) where thenumber of terms in the response function is considerably reduced. We define

Jn4n3n2n1ðs3; s2; s1Þ ¼

Z s2

0

ds00s

Z s00s

0

ds0s Cn4n3;n2n1ðs00s � s0sÞIn4ðs0sÞI�n3ðs3 � s0sÞIn2ðs2 � s00s ÞIn1ðs1 � s00s Þ; ðC:1Þ

with the eigenstate basis scattering matrix:

Cn4n3;n2n1ðsÞ ¼Xm4��m1

w�n4m4w�n3m3

Cm4m3;m2m1ðsÞwn2m2

wn1m1. ðC:2Þ

By transforming the coordinates to momentum space, we obtain from Eqs. (27)–(29):

SðIÞm4��m1ð�x4; . . . ;�x4Þ ¼ 2iXn4��n1

hdm4n4ðk4Þd

m�3n3ð�k3Þd

m�2n2ð�k2Þdm1

n1ðk1ÞiJn4n1n3n2

ðt41; t43; t42Þ; ðC:3Þ

SðIIÞm4��m1ð�x4; � � � ;�x4Þ ¼ 2iXn4��n1

hdm4n4ðk4Þd

m�3n3ð�k3Þdm2

n2ðk2Þdm1�

n1ð�k1ÞiJn4n2n3n1

ðt42; t43; t41Þ; ðC:4Þ

SðIIIÞm4��m1ð�x4; . . . ;�x4Þ ¼ 2iXn4��n1

hdm4n4ðk4Þdm3

n3ðk3Þdm2�

n2ð�k2Þdm1�

n1ð�k1ÞiJn4n2n3n1

ðt43; t42; t41Þ. ðC:5Þ

The wavevector-dependence enters through the transition dipoles. These expressions need to be rotationally averagedfor isotropic ensembles, as described in Appendix B.

Eqs. (C.3)–(C.5) can be conveniently expressed in terms of the frequency-domain scattering matrix (Eqs. (19)and (F.1)). These expressions then involve triple integrals (two with respect to time and one with respect to thescattering matrix frequency). However, the integration limits in Eq. (C.1) are controlled by multiple h(t) functionscoming from the Green�s functions. Taking these into account and using the frequency-domain scattering matrix


(Eq. (19)), the integrals over the exponential functions can be considerably simplified. The integration limits(�1,s2) for s00s and (0,s3) for s0s always hold for SI, SII and SIII in Eqs. (C.3)–(C.5). However, for S(I) (Eq.(C.3)) and SII (Eq. (C.4)) (but not for Eq. (C.5)) other integration limits (0,s2) for s0s can also be used. These allowto perform the time integrals:

SðIÞm4��m1ð�x4; . . . ;�x1Þ ¼ 2Xn4��n1

hdm4n4ðk4Þd

m�3n3ð�k3Þdm2�

n2ð�k2Þdm1

n1ðk1ÞiI�n1ðt21ÞI

�n1ðt32ÞIn2ðt32Þ

�Z

dx2p

Cn4n1;n3n2ðxÞIn3n2ðxÞIn4ðt43Þ � e�ixt43I�n1ðt43Þ

x� En4 � En1 þ iðcn4 � cn1Þ; ðC:6Þ

SðIIÞm4��m1ð�x4; . . . ;�x1Þ ¼ 2Xn4��n1

hdm4n4ðk4Þd

m�3n3ð�k3Þdm2

n2ðk2Þdm1�

n1ð�k1ÞiIn1ðt21ÞI�n2ðt32ÞIn1ðt32Þ

�Z

dx2p

Cn4n2;n3n1ðxÞIn3n1ðxÞIn4ðt43Þ � e�ixt43I�n2ðt43Þ

x� En4 � En2 þ iðcn4 � cn2ÞðC:7Þ

SðIIIÞm4��m1ð�x4; . . . ;�x1Þ ¼ 2Xn4��n1

hdm4n4ðk4Þdm3

n3ðk3Þdm2�

n2ð�k2Þdm1�

n1ð�k1ÞiIn1ðt21Þ

�Z

dx2p

Cn4n3;n2n1ðxÞIn2n1ðxÞe�ixt32 ½In4ðt43Þ � e�ixt43 I�n3ðt43Þ�x� En4 � En3 þ iðcn4 � cn3Þ

ðC:8Þ

Eqs. (C.6)–(C.8) constitute our most general expressions for the time domain nonlocal third-order responses inmomentum space. The wavevectors and times in these expressions correspond to the interaction events with the fields.All three response functions depend only on time delays between different interactions.

We next turn to the three signals defined by Eqs. (30)–(32). The polarization evolution is commonly transformed tothe frequency-domain where the signal spectra are displayed. The technique kI is known as photon echo. According tothe Feynman diagrams (Fig. 6) the system is transferred to a coherence by the first interaction. The second interactionleaves it either in a population or in a coherence between two excitonic states. We hold the second delay time, T2 (oftenreferred to as population or waiting time) fixed. Then the population and coherence evolution can be probed. The thirdinteraction creates coherences either between the ground and one-exciton states or between one- and two-excitonstates. To display the signal, we perform a double Fourier transform with respect to the first and third time delays:T1! X1 and T3! X3 in Eqs. (30)–(32). The signal also depends on the directions of optical wavevectors. Thus, theFourier transform of Eq. (30) can be performed analytically since the Green�s functions given by Eq. (15) are simpleexponential functions. We finally obtain,

PkIm4ðX3; T 2;X1Þ ¼

1

23expðþi/3 þ i/2 � i/1Þ

Xm4m2m1

SkIm4m3m2m1

ðX3; T 2;X1ÞEð3Þm3Eð2Þm2

Eð1Þm1. ðC:9Þ

In terms of the scattering matrix and the one-exciton Green�s functions, Eq. (C.6) gives:

SkIm4m3m2m1

ðX3; T 2;X1Þ ¼ 2iXn4��n1


n3ð�k3Þdm2�

n2ð�k2Þdm1

n1ð�k1ÞiI�n1ðT 2ÞIn2ðT 2ÞI�n1ð�X1ÞIn4ðX3Þ

�Z 1

�1

dx2p

Cn4n1;n3n2ðxÞIn3n2ðxÞI�n1ðx� X3Þ. ðC:10Þ

The frequency integration can be calculated analytically as follows. We note that CðxÞIðxÞ � iðx� 2�E þ 2i �cÞ�1 is atwo-exciton Green�s function with poles in negative imaginary half-plane (bars indicate averages), whileI�n1ðx� X3Þ ¼ �iðx� X3 � En1 � icn1Þ

�1 has one pole in positive imaginary half-plane:xp ¼ X3 þ En1 þ icn1 . In this case,we can use the Cauchy integral formula by closing the integration contour in the positive imaginary half-plane. Then

SkIm4m3m2m1

ðX3; T 2;X1Þ ¼ 2iXn4��n1


n3ð�k3Þdm2�

n2ð�k2Þdm1

n1ð�k1ÞiI�n1ðT 2ÞIn2ðT 2ÞI�n1ð�X1ÞIn4ðX3Þ

� Cn4n1;n3n2ðX3 þ En1 þ icn1ÞIn3n2ðX3 þ En1 þ icn1Þ. ðC:11Þ

Similar to kI, kII can be defined with the same Fourier transform and the same time delays:

PkIIm4ðX3; T 2;X1Þ ¼

1

23expðþi/3 � i/2 þ i/1Þ

Xm3m2m1

SkIIm4m3m2m1

ðX3; T 2;X1ÞEð3Þm3Eð2Þm2

Eð1Þm1. ðC:12Þ


In terms of the scattering matrix and the Green�s functions we get from Eq. (C.7):

SkIIm4m3m2m1

ðX3; T 2;X1Þ ¼ 2iXn4��n1


n3ð�k3Þdm2

n2ð�k2Þdm1�

n1ð�k1ÞiI�n2ðT 2ÞIn1ðT 2ÞIn1ðX1ÞIn4ðX3Þ

�Z 1

�1

dx2p

Cn4n2;n3n1ðxÞIn3n1ðxÞI�n2ðx� X3Þ ðC:13Þ

and after integration over frequency

SkIIm4m3m2m1

ðX3; T 2;X1Þ ¼ 2iXn4...n1


n3ð�k3Þdm2

n2ð�k2Þdm1�

n1ð�k1ÞiI�n2ðT 2ÞIn1ðT 2ÞIn1ðX1ÞIn4ðX3Þ

� Cn4n2;n3n1ðX3 þ En2 þ icn2ÞIn3n1ðX3 þ En2 þ icn2Þ. ðC:14Þ

This expression is very similar to kI.For kIII, two interactions with the delay T1 create a two-exciton coherence with k1 + k2. By performing the Fourier

transforms with respect to the second and third time delays: T2! X2 and T3! X3 we can observe the two-excitoncoherence along X2 and mixed, 0-to-one and one-to-two, coherences along X3:

PkIIIm4ðX3;X2; T 1Þ ¼

1

23expð�i/3 þ i/2 þ i/1Þ

Xm3m2m1

SkIIIm4m3m2m1

ðX3;X2; T 1ÞEð3Þm3Eð2Þm2

Eð1Þm1; ðC:15Þ

which is obtained from Eq. (C.8) as

SkIIIm4m3m2m1

ðX3;X2; T 1Þ ¼ 2iXn4��n1


n3ð�k3Þdm2�

n2ð�k2Þdm1�

n1ð�k1ÞiIn1ðT 1ÞIn4ðX3Þ

�Z 1

�1

dx2p

Cn4n3;n2n1ðxÞIn2n1ðxÞI�n3ðx� X3ÞhðX2 � xÞ. ðC:16Þ

The function h(x) = i(x + ic 0)�1 is taken in the limit cn� c 0 > 0. Analytic integration over frequency requires addi-tional terms. Again, CðxÞIðxÞ � iðx� 2�E þ 2i �cÞ�1 is a two-exciton Green�s function with poles in the negative imag-inary half-plane, while I�n1ðx� X3Þ ¼ �iðx� X3 � En1 � icn1Þ

�1 has one pole in positive imaginary half-plane:xp ¼ X3 þ En1 þ icn1 . h(X2 � x) = �i(x � X2 � ic 0)�1 has another pole with positive imaginary part: xp2 ¼ X2 þ ic0.In this case, the Cauchy integration results in two terms and

SkIIIm4m3m2m1

ðX3;X1; T 2Þ ¼ 2iXn4��n1


n3ð�k3Þdm2�

n2ð�k2Þdm1�

n1ð�k1ÞiIn1ðT 1ÞIn4ðX3ÞI�n3ðX2 � X3Þ

� Cn4n3;n2n1ðX2ÞIn2n1ðX2Þ � Cn4n3;n2n1ðX3 þ En3 þ icn3ÞIn2n1ðX3 þ En3 þ icn3Þ� �

; ðC:17Þ

where we have used cn� c 0.

Appendix D. The sequential pump–probe signal

Sequential pump–probe is an incoherent two-pulse FWM technique commonly used for monitoring excited statedynamics. The pump and the probe pulses are assumed well-separated and characterized by their carrier frequenciesxpu for pump and xpr for probe, and their delay spp. We ignore population transport and assume that the delay be-tween pump and probe shorter than the population evolution time. In addition, we assume that the pulse bandwidthsare much narrower than the exciton bandwidth (which always holds for electronic spectroscopy).

There are two interactions with the pump pulse (k1 = ±kpu, k2 = �kpu, where kpu is the pump pulse wavevector)and one with the probe (k3 = ±kpr, where kpr is the probe wavevector). We assume that the pump has various polar-ization components with phases /m

pu in Eq. (4), which can describe, e.g., circular polarization. Similarly, the probe canhave phases /m

pr. The pump–probe experiment measures the change in the probe absorption induced by the pump. Theabsorption of an optical field is given by [2,51]:

rA ¼Z

dxoPmðxÞ

otEmðxÞ

� �; ðD:1Þ

where Pm(x) is a third-order-induced polarization and Em(x) is a probe field when considering pump–probe experiment.Using the response function (Eq. (3)), we obtain for the pump–probe signal:


rPPðxpr; spp;xpuÞ ¼1

16

Xm4m3m2m1

Zdt4

Zdt3

Zdt2

Zdt1~E

ðprÞm4ðt4 � tprÞ~E

ðprÞm3ðt3 � tprÞ~E

ðpuÞm2ðt2 � tpuÞ~E

ðpuÞm1ðt1 � tpuÞ

�oSð3Þm4m3m2m1

ð�kprt4;þkprt3;þkput2;�kput1Þot4

exp½ixprðt4 � t3Þ � ixpuðt2 � t1Þ þ ið�/m4pr þ /m3

pr þ /m2pu � /m1

puÞ�(

þoSð3Þm4m3m2m1

ð�kprt4;þkprt3;�kput2;þkput1Þot4

exp½ixprðt4 � t3Þ þ ixpuðt2 � t1Þ þ ið�/m4pr þ /m3

pr � /m2pu þ /m1

puÞ�

þoSð3Þm4m3m2m1

ðþkprt4;�kprt3;þkput2;�kput1Þot4

exp½�ixprðt4 � t3Þ þ ixpuðt2 � t1Þ þ iðþ/m4pr � /m3

pr þ /m2pu � /m1

puÞ�

þoSð3Þm4m3m2m1

ðþkprt4;�kprt3;�kput2;þkput1Þot4

exp½�ixprðt4 � t3Þ þ ixpuðt2 � t1Þ þ iðþ/m4pr � /m3

pr � /m2pu þ /m1

puÞ�):

ðD:2Þ

Invoking the RWA and the other assumptions regarding the pulses made in Section 2, we find that this signal is a com-bination of the kI and kII techniques:

rPPðxpr; spp;xpuÞ ¼1

8xpr

Xm4m3m2m1

EðprÞm4EðprÞm4

EðpuÞm4EðpuÞm4

Im SkIm4m3m2m1

ðxpr; spp;�xpuÞeið�/m4prþ/

m3prþ/

m2pu�/

m1puÞ

n

þSkIIm4m3m2m1

ðxpr; spp;xpuÞeið�/m4prþ/

m3pr�/

m2puþ/

m1puÞo

ðD:3Þ

where kI = kpr + kpu � kpu and kII = kpr � kpu + kpu. It is possible to measure chiral components of this signal usingleft- and right-handed circular polarizations arranged such that non-chiral components cancel. When the pump band-width is larger than the exciton bandwidth Eq. (D.3) needs to be integrated over xpu.

Appendix E. Non-time-ordered response function and frequency-domain FWM signals

Using Eq. (24) we can define a non-time-ordered response function by:

Sm4��m1ðx4; . . . ; x1Þ ¼i

3

Xperm3

Xn4��n1

hMm4m3m2m1n4n3n2n1

ðr4; r3; r2; r1ÞiZ 1

�1ds00

Z 1

�1ds0

Xn04n03n02n01

Cn04n03;n0

2n01ðs00 � s0ÞGn4;n04

ðt4 � s00Þ

� Gyn03;n3ðs00 � t3ÞGn0

2;n2ðs0 � t2ÞGn0

1;n1ðs0 � t1Þ þ c.c.; ðE:1Þ

whereP

perm3denotes three terms in the following permutation: (m3r3t3,m2r2t2,m1r1t1), (m2r2t2,m3r3t3,m1r1t1) and

(m1r1t1,m2r2t2,m3r3t3). This form can be also expressed in the eigenstate basis using frequency domain scattering matrix.Eq. (E.1) can be substituted into Eq. (3) to give a non-time-ordered representation of the response. This form is very

convenient for frequency-domain four-wave-mixing processes [2]:

Pð3Þm4ðk4;x4Þ ¼

1

ð2pÞ12Xm3m2m1

Zdk3

Zdx3

Zdk2

Zdx2

Zdk1

Zdx1v

ð3Þm4;m3m2m1

ð�k4 � x4; k3x3; k2x2; k1x1Þ

� Em3ðk3;x3ÞEm2ðk2;x2ÞEm1ðk1;x1Þ; ðE:2Þ

where the susceptibility is obtained from Eq. (E.1) [17]:

vð3Þm4;m3m2m1ð�k4 � x4; k3;x3; k2;x2; k1;x1Þ ¼ 2pidðx4 � x3 � x2 � x1Þ

1

3

Xperm3

Xn4n3n2n1

hdm4n4ðk4Þdm3

n3ð�k3Þdm2�

n2ðk2Þdm1�

n1ðk1Þi

� Cn4n3;n2n1ðx2 þ x1ÞIn4ðx4ÞI�n3ð�x3ÞIn2ðx2ÞIn1ðx1Þ þ c.c.0. ðE:3Þ

whereP

perm3denotes a sum over the three permutations: (m3k3x3, m2k2x2, m1k1x1), (m2k2x2, m3k3x3, m1k1x1) and

(m1k1x1, m2k2x2, m3k3x3) (the expression is already symmetric to the permutation of m1k1x1 and m2k2x2).We consider CW laser fields with amplitudes EðsÞm , wavevectors ks and optical frequencies xs:

Emðk;xÞ ¼ð2pÞ4

2EðsÞm ei/

msdðkþ ksÞðx� xsÞ þ c.c.0 ðE:4Þ


For the signal in the direction �k � k2 + k3 we have:

Pð3Þm4ð�k1 � k2 þ k3;x1 þ x2 � x3Þ ¼

1

23

Xm3m2m1

e�i/m33þi/m2

2þi/m1

1 Eð3Þm3Eð2Þm2

Eð1Þm1

� vð3Þm4;m3m2m1ðk1 þ k2 � k3;�x1 � x2 þ x3; k3;�x3;�k2;x2;�k1;x1Þ; ðE:5Þ

In the RWA, this gives

vð3Þm4;m3m2m1ðk1 þ k2 � k3;�x1 � x2 þ x3; k3;�x3;�k2;x2;�k1;x1Þ

¼ 2pi1

3

Xn4n3n2n1

hdm4n4ð�k1 � k2 þ k3Þdm3

n3ð�k3Þdm2�

n2ð�k2Þdm1�

n1ð�k1ÞiCn4n3;n2n1ðx2 þ x1ÞIn4ðx1 þ x2 � x3Þ

� I�n3ðx3ÞIn2ðx2ÞIn1ðx1Þ þ c.c.0. ðE:6Þ

This is a three-dimensional signal of optical frequencies. Two dimensional x3 = x1 projections of this 3D signal havebeen calculated recently [17].

Appendix F. The exciton scattering matrix for periodic structures

In this appendix, we calculate the time-domain scattering matrix (Eq. (C.1)) for periodic systems. This can be easilyextended to non-periodic finite systems. Using Eq. (19), we calculate the scattering matrix in the frequency domain,which is directly used in Appendix C. The time-domain scattering matrix is then given by the Fourier transform:

Cn4n3;n2n1ðtÞ ¼Z

dx expð�ixtÞCn4n3;n2n1ðxÞ. ðF:1Þ

An infinite system is constructed by replicating a unit cell where all vibrational or electronic modes are fixed at par-ticular sites inside each cell in some crystal lattice. We will assume a cubic lattice of dimensionality D with a finite num-ber ND cells. To guarantee translational invariance, we use cyclic boundary conditions. Each mode is represented by apair of indices, Rm, where R is a position vector of the cell and m is the index of the site within the unit cell; the positionof the mth site inside the molecule is given by the vector R + qm, where R is the origin of the unit cell and qm is thedisplacement from that origin. We denote the number of sites in the cell by M and the lattice constant a. Since thesystem is translationally invariant, the intermode coupling JRm;R0n ¼ Jm;nðR0 � RÞ now depends on the distance betweencells R 0 � R and on the site indices of each cell, m and n. When R 0 = R, Jm, n(0), describes the coupling of modes insidethe cell. Jm,n(R

0 � R) with R 05R defines inter-cell couplings. Similar to this coupling the intermode anharmonicityDRm;R0n ¼ Dm;nðR0 � RÞ now depends on the distance between cells and on the site indices of each cell. We note thatDm,m(R

0�R) with R 0 5 R defines anharmonicity of the combination band, where two excitations are localized on dif-ferent sites.

The one-exciton eigenstates n of a periodic system are characterized by two quantum numbers: the Davydov bandindex k is related to different sites in the unit cell, and the exciton momentum (Bloch wavevector) q. For molecularsystems much smaller than the optical wavelength only zero momentum, q = 0, exciton states contribute to nonlinearoptical response. The infinite size only enters into the scattering matrix, where excitons with different momenta can beinvolved in the exciton scattering process.

The equations for the optical response (C.10)–(C.16) can be equivalently used for periodic structures with cyclicboundary conditions provided the eigenstates n are replaced with the periodic system eigenstates k at momentumq = 0. The scattering matrix is then considerably simplified:

Ck4k3;k2k1ðxÞ ¼Xm4��m1

0�wk4m4

�wk3m3

X�lc<r0r00<lc

�Cr00;m4;m3;r0;m2;m1ðxÞ

" #�wk2m2

�wk1m1; ðF:2Þ

where �Cr1;m;n;r2;m0;n0 is the mixed space scattering matrix (taken at zero momentum). This scattering matrix is given inmomentum space with respect to the translational motion of excitons, while it explicitly depends on real space coor-dinates r1 and r2, which are the distances between pairs of cells – these are not translationally invariant. This scatteringmatrix is given by [17]:

�Cr1;m;n;r2;m0;n0 ðxÞ ¼ �iDmnðr1Þð�DðxÞÞ�1r1;m;n;r2;m0 ;n0ðF:3Þ

and the matrix

�Dr1;m;n;r2;m0 ;n0 ðxÞ ¼ dr1;r2dm;m0dn;n0 þ i �gr1;m;n;r2;m0 ;n0

ðxÞDm0 ;n0 ðr2Þ; ðF:4Þ


where the mixed space two-exciton Green�s function

�gr1;m;n;r2;m0 ;n0

ðxÞ ¼ 1

m

Xq

e�iqðr2�r1Þgm;n;m0 ;n0 ðq;�q;xÞ ðF:5Þ

involves the sum over two-exciton Green�s function of one unit cell with different momenta:

gm;n;m0 ;n0 ðq;�q;xÞ;¼Xkk0

0�wkmðqÞ�wk0nð�qÞIkk0 ðq;�q;xÞ�w�km0 ðqÞ�w

�k0n0 ð�qÞ. ðF:6Þ

We have adopted the following notation. �wkm is a zero momentum wavefunction of exciton band k obtained fromP0m0Jm;m0 ðqÞ�wkm0 ðqÞ ¼ EkðqÞ�wkmðqÞ, where Jm;m0 ðqÞ ¼

PrJm;m0 ðrÞ expð�iqrÞ, and Ek(q) is the eigenenergy. The prime in

the sums denotes the summation either over sites within one cell or over different Davydov bands at zero momentum,while the sum over r is a sum over cells including r = 0 within the scattering length lc defined by Dmn(r > lc) = 0. All siteindices m run within one cell, while

Pq is the sum over momenta. For infinite systems, this sum becomes an integral

over all exciton bands.The scattering matrix calculation (Eq. (C.1)) is the same for finite non-periodic systems except that r and q are set to

0 in Eqs. (F.2)–(F.6), site indices m and n then correspond to different modes in the entire system and the exciton bandindices k should be changed to the exciton states n.

References

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time-domain chirally-sensitive three-pulse coherent probes ...the nee provide a uniﬁed treatment...

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