Quantum process tomography by 2D fluorescence spectroscopy

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Quantum Process Tomography by 2D Fluorescence Spectroscopy

Leonardo A Pachon1 2 Andrew H Marcus3 and Alan Aspuru-Guzik2

1Grupo de Fısica Atomica y Molecular Instituto de Fısica Facultad de Ciencias Exactas y Naturales

Universidad de Antioquia UdeA Calle 70 No 52-21 Medellın Colombia2Department of Chemistry and Chemical Biology Harvard University Cambridge MA 02138 USA

3Department of Chemistry Oregon Center for Optics Institute of Molecular Biology

University of Oregon Eugene Oregon 97403 United States

(Dated February 10 2015)

Reconstruction of the dynamics (quantum process tomography) of the single-exciton manifold inenergy transfer systems is proposed here on the basis of two-dimensional fluorescence spectroscopy(2D-FS) with phase-modulation The quantum-process-tomography protocol introduced here ben-efits from eg the sensitivity enhancement and signal-to-noise ratio ascribed to 2D-FS Althoughthe isotropically averaged spectroscopic signals depend on the quantum yield parameter Γ of thedoubly-excited-exciton manifold it is shown that the reconstruction of the dynamics is insensitiveto this parameter Applications to foundational and applied problems as well as further extensionsare discussed

PACS numbers 0365Yz 0570Ln 3710Jk

I INTRODUCTION

Since Chuang and Nielsenrsquo 1996 seminal proposal forexperimentally reconstruct the evolution operator of aquantum black box [1] a variety of proposals have beensuggested [2ndash4] and some experimental implementationshave been performed [5ndash7] Although most of these pro-posals have been suggested for relatively ldquocleanrdquo opticalsystems in the context of quantum information process-ing (QIP) recently quantum process tomography (QPT)met exciton dynamics [8ndash10] This closed a gap betweentheoretical and experimental studies on excitation energytransfer from the QIP and physical chemistry communi-ties

The current version of QPT for excitonic systems[8 9 11] is based on 2D Photon Echo Spectroscopy (2D-PES) [12 13] and therefore it relies on the wave-vectorphase-matching condition which works in extended sys-tems with many chromophores (see Refs [14 15] for de-tails) Hence the proposal developed in Refs [8ndash10]is not suitable for single-molecule QPT and resortingto eg phase-cycling (PCT) or phase-modulation tech-niques (PMT) [16 17] is desirable The reason for thisis that eg collinear PCT is not strongly limited bythe size of the sample and decreases the number of datapoints moreover it enhances the signal intensity by co-herently averaging specific interaction mechanisms (cfChap 7 in Ref [15])

Despite its advantages in proteins and molecular ag-gregates 2D electronic spectroscopy based on collinearPCT or PMT has been less commonly practiced thatfour-wave mixing approaches to 2D-PES However re-cent theoretical [17] and experimental [12 13] progresswith classical light have enabled PCT for a variety ofcomplex molecular systems relevant to exciton dynam-ics These recent developments and the general theoryof open quantum systemsndash quantum systems coupled tothe environmentndash are combined here to formulate a self-

consistent theory of QPT based on collinear PMT withsynchronous detectionFor the ideal situation when nonradiative processes

are neglected in the doubly-excited-exciton manifold thequantum yield parameter of this manifold is set to Γ = 2Under this circumstance it was shown that 2D FS coin-cides with 2D-PES [12 13 17] It is shown below thatthis equivalence also holds at the level of quantum pro-cess tomography ie the protocol introduced here gen-eralizes the protocol in Refs [8 9] to the more realisticsituation 0 le Γ lt 2

II INITIAL CONSIDERATIONS ON QPT

SYSTEM MODEL AND 2D-FS

Before introducing the reconstruction of the dynamicsit is necessary to point out some remarks on the basicsof process tomography the system-of-interest model and2D-FS

Quantum Process Tomography TensormdashIn quantummechanics the state of a physicochemical system S isdescribed by a density operator ρ Time evolution ofquantum states is governed by the Schrodinger equationwhich is linear in the state of the system This linearityallows for a description of the systemrsquos dynamics in termsof a linear map χt ρ0 7rarr ρt After projecting onto acomplete orthonormal basis |n〉 the map reads

〈n|ρ(t)|m〉 =sum

microν

χnmνmicro(t)〈ν|ρ(0)|micro〉 (1)

where χnmνmicro(t) stands for the process tomographytensor [2 8 18] For Hamiltonian dynamics with

H |n〉 = En|n〉 χnmνmicro(t) = eminusi(EmminusEn)t~δnνδmmicro [19ndash21] Thus population-to-coherence [χnmνν(t)] and thereverse [χnnνmicro(t)] process are prevented by the Kroneckerdeltas δnνδmmicro Clearly this restriction is not present if

2

driving fields are present or if the system of interest iscoupled to its environment [19ndash21]

In the general case of open quantum systems the func-tional form of Eq (1) remains valid under some condi-tions (i) If the coupling to the bath is weak Eq (1) holdsfor Markovian and non-Markovian processes and the pro-cess tensor is independent of the initial state (see egRefs [8 9] and references therein) (ii) If the coupling tothe bath is strong and initial system-environment corre-lations cannot be neglected Eq (1) holds after includingthose initial correlations in χnmνmicro(t) (see Refs [19ndash21]for details) (iii) Because initial bath correlations van-ish at high temperature even for strong coupling [22]then χnmνmicro(t) can be defined independently of the ini-tial state in the strong coupling regime entered at hightemperatures

After identifying the conditions under which Eq (1)holds it is relevant to consider some of the main proper-ties of the QPT tensor [8] namely

χnmνmicro = χlowastmnmicroν (2)

sum

n

χnnmicroν(T ) = δmicroν (3)

sum

nmνmicro

zlowastnνχnmνmicrozmmicro ge 0 (4)

where z is any complex valued vector Equation (2) en-sures the Hermitian character of the density operatorρ = ρdagger while Eq (3) guaranties probability conservationtrρ(t) = 1 The last property is a consequence of the factthat ρ(t) remains positive-semidefinite under unitary op-erations

The objective of QPT is the experimental reconstruc-tion of the process tomography tensor χnmνmicro(t)

ModelmdashConsider an excitonic dimer described by HS

and given by

HS = ω1adagger1a1 + ω2a

dagger2a2 + J

(

adagger1a2 + adagger

2a1

)

(5)

where adaggeri and ai are the creation and annihilation op-

erators for site i ω1 6= ω2 are the site energies whileJ 6= 0 is the Coulombic coupling between chromophoresBy defining the average frequency ω = 1

2 (ω1 + ω2)

the half-difference ∆ = 12 (ω1 minus ω2) and the mixing

angle θ = 12 arctan(J∆) it is possible to introduce

the creation and annihilation operators cp = cos θa1 +

sin θa2 and cdaggerp = sin θadagger1 + cos θadagger

2 of the p-th delocal-ized exciton state with energy ωp = ω plusmn ∆sec 2θ andp isin e eprime Starting from the ground state | g 〉 thesingle-exciton states are conveniently defined as | e 〉 =

cdaggere| g 〉 and | eprime 〉 = cdaggereprime | g 〉 while the biexciton state as

| f 〉 = adagger1a

dagger2| g 〉 = cdaggerec

daggereprime | g 〉 with ωf = ω1 + ω2 =

ωe + ωeprime The dipole vectors at each site are set tod1 = d1ez and d2 = d2 cos(φ)ez + d2 sin(φ)ex Sothat microeg = d2 sin θ sinφ ex + (d1 cos θ + d2 sin θ cosφ) ezmicroeprimeg = d2 cos θ sinφ ex + (minusd1 sin θ + d2 cos θ cosφ) ez

microfe = d2 cos θ sinφ ex + (d1 sin θ + d2 cos θ cosφ) ez andmicrofeprime = minusd2 sin θ sinφ ex + (d1 cos θ minus d2 sin θ cosφ) ez

Although exciton-exciton binding or repulsion termsare not included here it is considered that each excitonicmanifold contributes to the spectroscopic signal with aweight given by their fluorescence quantum yield coef-ficients Γν Specifically it is assumed that the quan-tum yield of the two singly excitonic states are the sameand equal to 10 while for the doubly excitonic mani-fold it is assumed that Γf = Γ with 0 le Γ le 2 Inthe ideal case in which two photons are emitted via thepath | f 〉 rarr | e eprime 〉 rarr | g 〉 Γ = 2 However because ofthe abundance of non-radiative relaxation pathways forhighly excited states the quantum yield of the doubly-excitonic manifold is expected to be smaller than that ofthe singly excitonic manifold in particular it is expectedthat Γ sim 0

For convenience the dimer Hamiltonian can be writ-ten as HS =

sum

ν=geeprimef ων |ν 〉〈 ν| To account for the

influence of the local vibrational environment in the exci-tonic dimer coupling to a thermally equilibrated phononbath at inverse temperature β is considered next Specif-ically the Hamiltonian of the environment is given by

HE =sum

p=eeprimesum

n ωnp

(

bdaggernpbnp + 12)

where ωnp de-

notes the frequency of the environment modes The in-teraction is described by HSE = Ee |e 〉〈 e|+ Eeprime |eprime 〉〈 eprime|+(

Ee + Eeprime

)

|eprime 〉〈 eprime| with Ep =sum

n λnp

(

bdaggernp + bnp

)

bdaggernp and bnp are the creation and annihilation bosonic op-erators of the nminusth mode of the vibrational environmentin the pminussite λnp measures the interaction strength be-tween the nminusth mode of the environment and the pminusthsite The net effect of the local environment is encodedin the spectral density Jn =

sum

n ω2npλ

2npδ(ω minus ωn)

2D Fluorescence Spectroscopy (2D-FS)mdashThe main dif-ference between the QPT scheme introduced below andprevious QPT proposals is the spectroscopic technique2D-FS that the present proposal is based on There-fore it is relevant to discuss the main differences andadvantages that 2D-FS has over eg 2D-PES In par-ticular 2D-FS makes use of the phase-modulation tech-nique and not of the wave-vector phase-matching alter-native [12 13 17] In contrast to phase-matching phase-modulation lock-in detection scheme used in 2D-FS canenhance the signal intensity and the signal-to-noise ra-tio by obtaining phase-sensitive spectroscopic informa-tion [16 17] Moreover the collinear character decreasesthe number of data points and in contrast to the phase-matching technique that works in extended systems withmany chromophores the collinear phase-modulation isnot strongly limited by the size of the sample [12 13 17]

The 2D-FS experiment monitors fluorescence which isproportional to the fourth-order excited populations

〈A(t)〉 = trAρ(4)(t) (6)

with A =sum

ν=eeprimef Γν |ν 〉〈 ν| generated by the action

3

of the operator V (tprime) that comprises the excitation byfour weak non-overlapping laser pulses

V (tprime) = minusλ4

sum

i=1

micromiddoteiE(tprimeminust)[

eminusiωi(tprimeminusti)+φi + cc

]

(7)

Here λ denotes the maximum intensity of the pulsesrsquo elec-tric field and micro the dipole operator ei ti ωi and φi

stand for the polarization vector time center frequencyand phase of the i-th laser pulse The pulse envelopeE(t) is chosen to be Gaussian with fixed width σ ie

E(t) = eminust22σ2

In the model under consideration theonly optically allowed transitions are between states dif-fering by one excitation Hence the only non-vanishingdipole transition matrix element are microij = microji withij = eg eprimeg fe feprime Details on the derivation and theexplicit functional form the fourth-order density matrixcan be found in the appendices A and BFor the purpose of extracting the QPT tensor from

the 2D-FS experimental signals only the rephasing sig-nals with global phase φreph = minusφ1 + φ2 + φ3 minus φ4 willbe considered below (see Fig 1) Thus assuming thatthe rotating wave approximation (RWA) holds the inter-actions with the electromagnetic fields are characterizedby

V1 = minusλmicrolt middot e1E(tminus t1)eiω1(tminust1) (8)

V2 = minusλmicrogt middot e2E(tminus t2)eminusiω2(tminust2) (9)

V3 = minusλmicrogt middot e2E(tminus t3)eminusiω3(tminust3) (10)

V4 = minusλmicrolt middot e4E(tminus t4)eiω4(tminust4) (11)

where microlt =

sum

ωpltωqmicropq|p〉〈q| promotes emissions from

the ket and absorptions on the bra and microgt = (microlt)dagger in-

duces the opposite processes For this particular selectionof the global phase φreph = minusφ1 + φ2 + φ3 minus φ4 the 2D-FS signals are equivalent to the rephasing spectroscopicsignals in the photon-echo direction kPE = minusk1+k2+k3

when Γ = 2 [12 13 17] Below it is shown thatthe present QPT protocol reduces to the protocol inRefs [8 9] when Γ = 2 as well

III 2D-FS QPT

As stated above the main goal of QPT is the recon-struction of the dynamics of the density operator In do-ing so it is assumed that the structural parameters of themodel namely the transition frequencies ωij = ωi minus ωj

and the dipole transition matrix element microij are allknown This pre-requisite is not an issue because in-formation about the transition frequencies is routinelyobtained from linear absorption spectra and the dipolescan be inferred form X-ray crystallography [8]Once the structural parameters are given the recon-

struction of the dynamics comprises three main parts(i) initial state preparation (ii) evolution and (iii) final

state detection In describing these stages it is usefulto introduce the standard time intervals τ T t insteadof the time center ti of each pulse [14 15] The timedifference between the second and the first pulse definesthe coherence time interval t = t2 minus t1 The time inter-val between the third and the second pulse T = t3 minus t2is known as the waiting time and defines the quantumchannel to be characterized by the QPT scheme Finallythe difference between the fourth and the third pulset = t4 minus t3 denotes the echo timeInitial State PreparationmdashThe excitonic system be-

fore any electromagnetic perturbation is assumed to bein the ground state ρ(minusinfin) = |g 〉〈 g| Thus the basicidea is to make use of the first two pulses to prepare theeffective initial density matrix at T = 0 ρω1ω2

e1e2(T = 0)

and use the last two pulses to read out the stateAfter applying second order perturbation theory in λ

and under the assumption that RWA holds in this case(see Appendix A for details) the effective initial statereads

ρω1ω2

e1e2(0) =minus

sum

pqisineeprime

Cpω1Cq

ω2(micropg middot e1)(microqg middot e2)

times Ggp(τ) (|q〉〈p| minus δpq|g〉〈g|) (12)

where Gij(τ) is the propagator of the optical coherence|i 〉〈 j| For simplicity it can be assumed as Gij(τ) =Θ(τ) exp[(minusiωij minus Γij)τ ] begin Γij dephasing rates andthe Heaviside function Θ(τ) ensures causality The co-efficients Cp

ωiare purely imaginary and given by Cp

ωi=

iλradic2πσ2eminusσ2(ωpgminusωi) Because at this level there is no

influence of the doubly-excited exciton manifold the ef-fective initial state in Eq (12) coincides with the effectiveinitial state prepared by 2D-PES in Ref [8]

The contributions to ρω1ω2

e1e2(0) in Eq (12) are clearly

depicted in the lower panel of Fig 1 (see Figs A B Cand D) Starting from the system the ground state |g 〉〈 g|in the RWA and selecting only those contributions withminusφ1+φ2 the first pulse can only excite the bra and thencreates an the optical coherence |g 〉〈 p| with probabilityCp

ω1 This coherence |g 〉〈 p| evolves under the action of

Ggp(τ) for a time τ when the second pulse prepares thestate |q 〉〈 p| with probability Cq

ω2or a hole minus |g 〉〈 g| with

probability Cpω2

As in the case of QPT based on 2D-PES [8 9] to pre-pare the set of four linearly independent states in Eq (12)(see also Figs 1Andash1D) it suffices to consider a pulse tool-box of two waveforms with carrier frequencies ω+ ωminusthat create | e 〉 and | eprime 〉 with different amplitudes Ofcourse the discrimination in the preparation of | e 〉 and| eprime 〉 depends on how close to resonance the carrier fre-quencies are For an extensive and detailed analysis onthis respect see Refs [8 9]EvolutionmdashOnce the initial state ρω1ω2

e1e2(0) is effec-

tively prepared ie after the action of the first-two

4

FIG 1 Double-sided Feynmanrsquos diagrams for the initial state preparation (lower panel) and state detection (upper panel)that lead to the rephasing spectroscopic signals [SFS]

ω1ω2ω3ω4e1e2e3e4

(τ T t) in Eq (14) The 2D-FS signals are synchronously-phasedetected with respect to the modulated laser fields at frequency φrep = minusφ1 +φ2 +φ3 minusφ4 In the lower and upper panels thediagrams are grouped according to the probability they occur Cp

ω1Cq

ω2and Cp

ω3Cq

ω4 respectively

pulses T amp 3σ the system evolves over a time T un-der the action of the super operator χ(T ) according to

ρω1ω2

e1e2(T ) = χ(T )ρω1ω2

e1e2(0) (13)

To avoid contamination of the initial state by terms pro-portional to a hole every time there is a single-excitonpopulation |p 〉〈 p| it is assumed that 〈ab|χ(T )|gg〉 =χabgg(T ) = δagδbg which is equivalent to neglect pro-cesses where phonons induce upward optical transitionsand spontaneous excitation from the single to the dou-ble exciton manifolds [8] Up to this condition χ(T ) inEq (13) describes the dynamics induced by any bathmodel and accounts for any system-bath couplingFinal State DetectionmdashThe very nature of the fluores-

cence detection in 2D-FS suggests considering contribu-tions from excitation configurations that lead to popula-tions only In the upper panel of Fig 1 those contribu-tions are schematically displayed and grouped accordingto the probability Cp

ω3Cq

ω4they occur

In contrast to QPT based on 2D-PES there are herefourteen possibilities for the final state instead of tenthus twenty independent experiments are needed insteadof sixteen This comes at the expense of the different rolethat the fourth pulse has in each technique namely het-erodyne detection in 2D-PES and the generation of pop-ulations in 2D-FS However the signals that lead to pop-

ulation of the doubly-excited exciton manifold | f 〉〈 f |from the coherence | f 〉〈 e | must be summed up to thesignal that lead to the population | e 〉〈 e | In the sum-mation the process | f 〉〈 e | rarr | f 〉〈 f | is weightedwith a factor Γ while the process | f 〉〈 e | rarr | e 〉〈 e |has weight -1 The same procure applies to the process| f 〉〈 eprime | rarr | f 〉〈 f | and | f 〉〈 eprime | rarr | eprime 〉〈 eprime | This pro-cedure leads to the sixteen independent signals that areneeded to reconstruct the sixteen elements of the processtensor χnmνmicro(T )

By using the same toolbox as in the preparation stateand following closely the notation in Ref [8] the totalsignal is given by

[SFS]ω1ω2ω3ω4

e1e2e3e4(τ T t)

=sum

pqrsisineeprime

Cpω1Cq

ω2Cr

ω3Cs

ω4P pqrse1e2e3e4

(14)

with

P pqeee1e2e3e4

(τ T t) =

(micropg middot e1) (microqg middot e2)Ggp(τ)

times (microeg middot e3) (microeg middot e4)Geg(t)

times [χqqqp(T )minus δpq minus χeeqp(T )]minus (1minus Γ)

times (microfeprime middot e3) (microfeprime middot e4)Geg(t)χeprimeeprimeqp(T )

(15)

5

and

P pqeeprime

e1e2e3e4(τ T t) =

minus (micropg middot e1) (microqg middot e2)Ggp(τ)

times (microeg middot e3) (microeprimeg middot e4)Geprimeg(t)χeprimeeqp(T )

+ (1 minus Γ) (microfeprime middot e3) (microfe middot e4)Gfe(t)

(16)

Analogous expressions hold for P pqeprimeeprime

e1e2e3e4and

P pqeprimeee1e2e3e4

after interchanging e harr eprime The 2D-FSsignals [SFS]

ω1ω2ω3ω4

e1e2e3e4(τ T t) in Eq (14) with (15) and

(16) are the main result of this article They allow forthe reconstruction of the dynamics of excitonic systemsbased on 2D-FS that is an attractive approach to reachQPT at the level of single molecules Remarkably theappealing form of Eqs (14) (15) and (16) allows foran immediate connection with the protocol derived inRefs [8 9] Specifically up to a global minus sign thatis consistent with previous investigations [12 13] resultsin Refs [8 9] are obtained by simply setting Γ = 2 inEqs (14) (15) and (16)

Because the probed sample is an ensemble of isotropi-cally distributed molecules in solution an isotropic aver-age of (microa middot e1) (microb middot e2) (microc middot e3) (microd middot e4) is needed Indoing so standard procedures are followed (see egChap 11 in Ref [23] or Sec 33 in Ref 15) Specificallythe isotropic average is given by

〈(microa middot e1) (microb middot e2) (microc middot e3) (microd middot e4)〉iso=

sum

m1m2m3m4

I(4)e1e2e3e4m1m2m3m4

times (microa middotm1) (microb middotm2) (microc middotm3) (microd middotm4)

(17)

where ei and mi denote the polarization of the pulsesin the laboratory and molecule-fixed frames respectivelyei = exi eyi exi and mi = mximyimxi are thecomponents of the polarization vectors ei and mi re-spectively The isotropically invariant tensor I(4) is givenby

I(4)e1e2e3e4m1m2m3m4

=

1

30(δe1e2δe3e4 δe1e3δe2e4 δe1e4δe2e3)

times

4 minus1 minus1minus1 4 minus1minus1 minus1 4

δm1m2δm3m4

δm1m3δm2m4

δm1m4δm2m3

(18)

The explicit expression for the relevant case of interestin the collinear configuration used in 2D-FS e1 = e2 =e3 = e4 = z can be found in Appendix C Thus afterisotropically averaging

〈[SFS]ω1ω2ω3ω4

e1e2e3e4(τ T t)〉iso

=sum

pqrsisineeprime

Cpω1Cq

ω2Cr

ω3Cs

ω4〈P pqrs

e1e2e3e4〉iso (19)

The extraction procedure of the matrix elements of χ

from 〈[SFS]ω1ω2ω3ω4

e1e2e3e4(τ T t)〉iso follows from Eqs (19)

(15) and (16) In doing so note that the sixteen 2D-FS signals 〈[SFS]

ω1ω2ω3ω4

e1e2e3e4(τ T t)〉iso and the sixteen

auxiliary signals 〈P pqrse1e2e3e4

(τ T t)〉iso can be groupedin the sixteen-dimensional vectors 〈[SFS](τ T t)〉isoand 〈P(τ T t)〉iso respectively This allows writingEq (19) as 〈[SFS](τ T t)〉iso = C〈P(τ T t)〉iso wherethe matrix elements of C contains the probabilitiesCp

ω1Cq

ω2Cr

ω3Cs

ω4 Then the first step in the extraction

procedure is to invert the matrix C so that the sig-nals 〈P pqrs

e1e2e3e4〉iso can be extracted from the measured

signals 〈[SFS]ω1ω2ω3ω4

e1e2e3e4(τ T t)〉iso ie 〈P(τ T t)〉iso =

Cminus1〈[SFS](τ T t)〉iso The second step comprises the ex-traction of the sixteen elements of the process tensorχ(T ) from the isotropically-averaged version of Eqs (15)and (16) This process can be accomplished by conve-niently defining a sixteen-dimensional vector χ(T ) suchthat χ(T ) = Mminus1〈P(0 T 0)〉iso See the Appendices forfurther details

IV NUMERICAL EXAMPLE

As a concrete example consider parameters of rele-vance in the context of light-harvesting systems [24 25]Specifically to compare with previous results [8] considerω1 = 12881 cmminus1 ω2 = 12719 cmminus1 J = 120 cmminus1d2d1 = 2 and φ = 03 The two-waveform toolboxis assumed to have frequencies ω+ = 13480 cmminus1 andωminus = 12130 cmminus1 so that ωi = ω+ ωminus forall i and pulsewidth σ = 40 fs To simulate the signals the spectraldensity Jn(ω) of the local vibrational environments areassumed identical and given by (λreωc)ω exp(minusωωc)where the cutoff frequency is set as ωc = 120 cmminus1 whilethe reorganization energy is chosen as λre = 30 cmminus1These set of parameters are relevant for light-harvestingsystems and were used in Ref [8]In the simulations below an inhomogeneously broad-

ened ensemble of 104 dimers with diagonal disorder isconsidered Specifically it is assumed that the site en-ergies ωprime

1 and ωprime2 in the ensemble follow a Gaussian dis-

tribution centered at ω1 and ω2 with standard deviationσinh = 40 cmminus1 The dynamics are solved at the levelof the secular Redfield master equation at room temper-ature and for T ge 3σFigure 2 depicts the nonvanishing real parts of

〈P pqrszzzz〉iso for a variety of values of the quantum yields

parameter Γ From the functional dependence on Γ ofthe signals 〈P pqrs

zzzz 〉iso in Eqs (15) and (16) three casesare of interest (i) For Γ = 0 the contribution from the ex-cited state absorption (ESA) pathways has the same signas the stimulated emission (SE) and ground-state bleach(GSB) contributions (see the double-sided Feynmanrsquos di-agrams in Fig 1 or the discussion in Ref [13]) Thus theamplitude of the signals is the largest possible (ii) ForΓ = 1 the ESA pathways do not contribute to the signaland the amplitude of the signals is expected to be smallerthan in the case Γ = 0 (iii) For Γ = 2 the contribution

6

minus40

minus20

0

lang Pee

eezzzz(t)rang

Γ=00 Γ=05 Γ=10 Γ=15 Γ=20

minus40

minus20

0

20

40

lang Pee

eprimeeprime

zzzz(t)rang

minus20

minus10

0

lang Peprime

eprimeee

zzzz(t)rang

minus20

minus10

0

10

20

lang Peprime

eprimeeprime

eprimezzzz

(t)rang

minus10

0

10

lang Peprime

eeprime e

zzzz(t)rang

100 200 300 400 500 600 [fs]minus20

minus10

0

10

20

lang Pee

prime eeprime

zzzz(t)rang

FIG 2 Nonvanishing 〈P pqrs

zzzz (0 T 0)〉iso signals for a varietyof values of Γ and for 3σ le T le 700 fs Curves with Γ = 2coincides with those extracted in Ref [8]

from the ESA pathways has opposite sign to the SE andGSB contributions so that amplitude is expected to befurther small than in the previous case Γ = 1 Theseexpectations are confirmed by simulations in Fig 2 Forcompleteness the intermediate cases Γ = 05 and Γ = 15were also depicted in Fig 2

Based on the signals 〈P pqrszzzz〉iso obtained above the

QPT tensor is reconstructed in Fig 3 The reconstruc-tion seems to be insensitive to the value of the quantumyield parameter Γ This unexpected result can be un-derstood after noticing that the QPT tensor χnmνmicro(T )is a characteristic of the singly exited exciton manifoldand Γ is a function of the doubly-excited exciton man-

ifold Thus QPT of the singly-exited exciton manifoldby 2D-FS is robust against nonradiative processes of thedoubly-excited exciton manifold and benefits from thequality of the signals discussed above

minus10

0

10

χee

ee(t)

Γ=00 Γ=05 Γ=10 Γ=15 Γ=20

minus10

0

10

χee

eprimeeprime(t)

minus10

0

10

χeprime

eprimeee(t)

minus10

0

10χ

eprimeeprime

eprimeeprime(t)

minus10

0

10

realχee

prime eeprime(t)

100 200 300 400 500 600 [fs]minus10

0

10

imageχee

prime eeprime(t)

FIG 3 Nonvanishing elements of the QPT tensor χnmνmicro(T )obtained form the 〈P pqrs

zzzz 〉iso depicted in Fig 2 for a varietyof values of Γ and for 3σ le T le 700 fs Curves with Γ = 2coincides with those extracted in Ref [8]

Results in Fig 3 agree with the secular Redfield tensorused to simulate the 2D-FS signals If the signal wereobtained from experimental data a careful analysis ofthe propagation of errors is in order In particular itis necessary to include fluctuations in the laser intensityat each time T at which the signals are collected and topay attention to the stability conditions imposed by theinvertibility of the matrices C and M [8] In this arti-cle interest was in providing a proof of principle for the

7

scheme derived above so that Fig 3 is aimed to depictthe type of information that can be extracted from theprotocol

Specifically (i) if for a particular photochemical sys-tem non-negligible non-secular terms emerge during thereconstruction of the process tensor χ from experimen-tal data that would imply eg that coherent controlschemes assisted by the environment [20 26] may be ap-plied in that particular system (ii) If the decay of thetensor elements associated to the coherences of the den-sity matrix χnmnm with n 6= m differs from exponencialit may indicate the presence of non-Markovian dynamics[27] The deviation from exponencial decay may be evenconsidered as a measure of the non-Markovian characterof the dynamicsndash a relevant topic in the context of openquantum systems (iii) Although in multilevel systemsthe decay rate of the elements χnmnm cannot be directlyassociated to the decay rate of the coherences 〈n|ρ|m〉of the density matrix the decay rate of χnmnm providesinformation on the lifetime of particular transfer and co-herent mechanisms

V DISCUSSION

Having experimental access to the process tomographytensor χnmνmicro(t) is fundamental in eg revealing energypathways in exciton dynamics and in designing controlstrategies to increase transport efficiency Specificallyapplications of QPT to photosynthetic light-harvestingsystems can eg (i) rule out certain transfer mecha-nisms proposed in the literature and (ii) address the ques-tion about the quantumclassical nature of the energytransport in these biological systems from an experimen-tal viewpoint In addressing these respects a completeanalysis of the classicalquantum correlations encoded inthe process tomography tensor as well as an analysis ofthe main contributing elements to energy transport isrequired and will be discussed elsewhere

A variety of applied and foundational problems canbe addressed once the process tomography tensor is re-constructed From a foundational viewpoint if the pro-cess tomography tensor is translated into the phase-spacerepresentation of quantum mechanics it reduces to thepropagator of the Wigner function [28ndash30] Based on thisobject it is possible to experimentally reconstruct signa-tures of quantum chaos such as scars with sub-Planckianresolution [28] Phase-space resolution below ~ can beachieved here because the process tomography tensor or

equivalently the propagator of the Wigner function isnot a physical state and therefore it is not restricted bythe uncertainty principle [28]In the same way that 2D-PES was extended to study

chemical exchange to obtain reaction rates under wellcontrolled conditions (see eg Chap 10 in Ref 15) astraightforward extension of QPT is the accurate mea-surement of concentration of different species in chem-ical reactions This has been considered very recentlyin the literature [31] In this context interest is in thepopulation dynamics χnnνν(T ) of the different chemicalspecies that under Markovian dynamics are in accor-dance to detailed balance and the Onsagerrsquos regressionhypothesis [22 32 33] In this respect because ultrafastspectroscopy allows for the study of chemical exchangewith no need of pressure temperature pH nor concen-tration jumps it is expected that the proposed approachprovides experimental evidence for the failure of the On-sagerrsquos regression hypothesis induced by non-Markoviandynamics at the quantum level [22 32 33]The QPT scheme introduced here can be readily im-

plemented at the experimente level and constitutes afirst step toward the formulation of QPT at the single-molecule level Such a scheme would certainly incor-porate quantum aspects of the electromagnetic radia-tion such as the use of energy-entangled photons [34]This is is already under development in our laborato-ries Finally based on present non-linear optical activityspectroscopy (see eg Chap 16 in Ref 15) by intro-ducing polarization-entangled photons instead of energy-entangled photons [34] single-molecule QPT may be ex-tended to study optical active materials at the single-molecule level These materials exhibit unique opticalproperties and are constantly finding applications in sci-ence and industry

ACKNOWLEDGMENTS

Discussions with Keith Nelson and Joel Yuen-Zhou areacknowledged with pleasure This work was supported bythe Center for Excitonics an Energy Frontier ResearchCenter funded by the US Department of Energy Of-fice of Science and Office of Basic Energy Sciences underAward Number DE-SC0001088 by Comite para el De-sarrollo de la Investigacion ndashCODIndash of Universidad deAntioquia Colombia under the Estrategia de Sostenibil-idad 2015-2016 and by the Colombian Institute for theScience and Technology Development ndashCOLCIENCIASndashunder the contract number 111556934912

[1] I L Chuang and M A Nielsen J Mod Opt 44 2455(1997)

[2] M Mohseni and D A Lidar Phys Rev Lett 97 170501(2006)

[3] A Bendersky F Pastawski and J P Paz Phys RevLett 100 190403 (2008)

[4] A Shabani R L Kosut M Mohseni H Rabitz M ABroome M P Almeida A Fedrizzi and A G White

8

Phys Rev Lett 106 100401 (2011)[5] Z-W Wang Y-S Zhang Y-F Huang X-F Ren and

G-C Guo Phys Rev A 75 044304 (2007)[6] C T Schmiegelow M A Larotonda and J P Paz

Phys Rev Lett 104 123601 (2010)[7] D Nigg J T Barreiro P Schindler M Mohseni

T Monz M Chwalla M Hennrich and R Blatt PhysRev Lett 110 060403 (2013)

[8] J Yuen-Zhou J J Krich M Mohseni and A Aspuru-Guzik Proc Natl Acad Sci USA 108 17615 (2011)

[9] J Yuen Zhou and A Aspuru Guzik J Chem Phys 134134505 (2011)

[10] J Yuen-Zhou D H Arias D M Eisele C P SteinerJ J Krich M G Bawendi K A Nelson and A Aspuru-Guzik ACS Nano 8 5527 (2014)

[11] J Yuen-Zhou J J Krich I Kassal A S Johnson andA Aspuru-Guzik in Ultrafast Spectroscopy 2053-2563(IOP Publishing 2014) pp 1ndash1 to 1ndash9

[12] G A Lott A Perdomo-Ortiz J K Utterback J RWidom A Aspuru-Guzik and A H Marcus Proc NatlAcad Sci USA 108 16521 (2011)

[13] A Perdomo-Ortiz J R Widom G A Lott A Aspuru-Guzik and A H Marcus J Phys Chem B 116 10757(2012)

[14] S Mukamel Principles of Nonlinear Optical Spectroscopy

(Oxford University Press 1999)[15] M Cho Two-Dimensional Optical Spectroscopy (Taylor

amp Francis 2010)[16] P F Tekavec T R Dyke and A H Marcus J Chem

Phys 125 194303 (2006)[17] P F Tekavec G A Lott and A H Marcus J Chem

Phys 127 214307 (2007)[18] M-D Choi Linear Algebra Appl 10 285 (1975)

[19] L A Pachon and P Brumer Phys Rev A 87 022106(2013) arXiv12106374 [quant-ph]

[20] L A Pachon and P Brumer J Chem Phys 139 164123(2013) arXiv13081843

[21] L A Pachon and P Brumer J Math Phys 55 012103(2014) arXiv12073104

[22] L A Pachon J F Triana D Zueco and P BrumerarXiv14011418

[23] D Craig and T Thirunamachandran Molecular Quan-

tum Electrodynamics Dover Books on Chemistry (DoverPublications 2012)

[24] P Kjellberg B Bruggemann and T Pullerits PhysRev B 74 024303 (2006)

[25] H Lee Y-C Cheng and G R Fleming Science 3161462 (2007)

[26] L A Pachon L Yu and P Brumer Farad Discuss163 485 (2013) arXiv12126416

[27] L A Pachon and P Brumer J Chem Phys 141 174102(2014) arXiv14104146 [quant-ph]

[28] T Dittrich and L A Pachon Phys Rev Lett 102150401 (2009) arXiv08113017

[29] T Dittrich E A Gomez and L A Pachon J ChemPhys 132 214102 (2010) arXiv09113871

[30] L A Pachon G-L Ingold and T Dittrich Chem Phys375 209 (2010) arXiv10053839

[31] A Crespi M Lobino J C F Matthews A Politi C RNeal R Ramponi R Osellame and J L OrsquoBrien AppPhys Lett 100 233704 (2012)

[32] P Talkner Ann Phys 167 390 (1986)[33] G W Ford and R F OrsquoConnell Phys Rev Lett 77

798 (1996)[34] M G Raymer A H Marcus J R Widom and D L P

Vitullo J Phys Chem B 117 15559 (2013)

Appendix A Initial State Preparation

Because the effective initial state ρω1ω2

e1e1(t+ T ) is prepared by the first two pulses it is of second order in λ Thus

after applying second order perturbation theory to the time evolution of the system density matrix (see eg Chap 5in Ref [14]) the effective initial state reads

ρω1ω2

e1e1(t+ T ) =

(

1

i

)2 int t2+T

minusinfin

dtprimeprimeint tprimeprime

minusinfin

dtprimeG2(t2 + T tprimeprime)V (tprimeprime)G1(tprimeprime tprime)V (tprime)|g〉〈g| (A1)

where it was assumed that ρ(minusinfin) = |g〉〈g| Symbols in calligraphic font denote superoperators in particular

V =sum4

i=1 Vi(t) where Vi = [Vi middot] Assuming that the rotating wave approximation (RWA) holds and that therephasing signal is synchronously-phase detected at φ = minusφ1 +φ2 +φ3 minusφ4 the interaction with the electromagneticradiation is conveniently described by

V1 = minusλmicrolt middot e1E(tminus t1)eiω1(tminust1) (A2)

V2 = minusλmicrogt middot e2E(tminus t2)eminusiω2(tminust2) (A3)

V3 = minusλmicrogt middot e2E(tminus t3)eminusiω3(tminust3) (A4)

V4 = minusλmicrolt middot e4E(tminus t4)eiω4(tminust4) (A5)

where microlt =

sum

ωpltωqmicropq|p〉〈q| promotes emissions from the ket and absorptions on the bra and micro

gt = (microlt)dagger induces

the opposite processes

In the following it is considered that the first and the second pulse are well separated ie τ = t2 minus t1 gt 3σ This

9

allows for the substitutions V (tprimeprime) = V2(tprimeprime) and V (tprimeprime) = V1(t

prime)

ρω1ω2

e1e1(t+ T ) = λ2

sum

pq

t2+Tint

minusinfin

dtprimeprimeχ(T )

Gqp(t2 minus tprimeprime)(

microqg middot e2 |q 〉〈 g|E(tprimeprime minus t2)eiω1(t

primeprimeminust2))

timestprimeprimeint

minusinfin

dtprimeGgp(tprimeprime minus tprime) |g 〉〈 g|

(

microqg middot e2 |q 〉〈 g|E(tprimeprime minus t2)eminusiω2(t

primeprimeminust2))

minust2+Tint

minusinfin

dtprimeprimeχ(T )

Ggg(t2 minus tprimeprime)

tprimeprimeint

minusinfin

dtprimeGgp(tprimeprime minus tprime) |g 〉〈 g|

(

micropg middot e1 |q 〉〈 g|E(tprime minus t1)eiω1(t

primeminust1))

times(

microqg middot e1 |q 〉〈 g|E(tprimeprime minus t2)eminusiω2(t

primeprimeminust2))

(A6)

If the duration of the pulse σ is much sorter than the dynamics induced by the environment characterized by Γnmie if σ ≪ Γminus1

nm then decoherering contributions can be neglected so that Ggp(t1 minus tprime) asymp exp [iωpg(t1 minus tprime)] Gqp(t2 minustprimeprime)Ggp(t

primeprime minus t2) asymp exp [minusiωqp(t2 minus tprimeprime)] exp [iωpg(t2 minus tprimeprime)] = exp [minusiωqg(t2 minus tprimeprime)] Ggp(tprimeprime minus t2) asymp exp [minusiωpg(t

primeprime minus t2)]and Ggg(t

primeprime minus t2) asymp exp [minusiωgg(tprimeprime minus t2)] Moreover Ggp(t

primeprime minus tprime) asymp Ggp(tprimeprime minus t2)Ggp(t1 minus tprime)Ggp(t

prime2 minus t1) After some

manipulations

ρω1ω2

e1e1(t+ T ) asymp minusχ(T )

sum

pq

Cpω1Cq

ω2(micropg middot e1)(microqg middot e2)Ggp(τ) (|q 〉〈 p| minus δpq |g 〉〈 g|)

(A7)

with

Cpωj

= minusλ

i

int infin

minusinfin

ds exp[i(ωj minus ωpg)s]E(s) = minusλ

i

radic2πσ2 exp[minus(ωpg minus ωj)

2] (A8)

As mentioned in the main text this effective initial state coincides with the one prepared by 2D-PES in Ref [8]

Appendix B Final State Detection

To derive the explicit form of the density operator after the action of the four pulses it is assumed that the thirdand the fourth pulses are well separated as well To account for the action of the third pulse and a subsequent periodof free evolution perturbation theory is applied once more so that the density operator of the system reads

ρω1ω2ω3

e1e2e3(τ + T + t) =

sum

pq

[

Ceω3(microeg middot e3)Geg(t)〈g

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ g〉

minusCeω3(microeg middot e3)Geg(t)〈e

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ e〉 minus Ceprime

ω3(microeprimeg middot e3)Geprimeg(t)〈e

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ eprime〉]

|e 〉〈 g|

+[

Ceprime

ω3(microeprimeg middot e3)Geprimeg(t)〈g

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ g〉

minusCeprime

ω3(microeprimeg middot e3)Geprimeg(t)〈eprime

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ eprime〉 minus Ceω3(microeg middot e3)Geprimeg(t)〈eprime

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ e〉]

|eprime 〉〈 g|

+[

Ceprime

ω3(microfe middot e3)Gfe(t)〈e

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ e〉+ Ceω3(microfeprime middot e3)Gfe(t)〈eprime

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ e〉]

|f 〉〈 e|

+[

Ceprime

ω3(microfe middot e3)Gfeprime(t)〈e

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ eprime〉+ Ceω3(microfeprime middot e3)Gfeprime(t)〈eprime

∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ eprime〉]

|f 〉〈 eprime|

(B1)

where

〈i∣

∣ρω1ω2

e1e1(t+ T )

∣

∣ j〉 = minussum

pq

Cpω1Cq

ω2(micropg middot e1)(microqg middot e2)Ggp(τ) (χijqp(T )minus δpqδijδig) (B2)

10

Finally the fourth pulse prepares the system in the state

ρω1ω2ω3ω4

e1e2e3e4(τ + T + t)

=(

Ceprime

ω4(microef middot e4)〈f

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ e〉 minus Ceω4(microge middot e4)〈g

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ e〉)

|e 〉〈 e|

+(

Ceω4(microeprimef middot e4)〈f

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ eprime〉 minus Ceω4(microge middot e4)〈eprime

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ g〉)

|eprime 〉〈 eprime|

minus(

Ceprime

ω4(microeeprime middot e4)〈f

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ e〉+ Ceω4(microeprimef middot e4)〈f

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ eprime〉)

|f 〉〈 f |

(B3)

Each contribution can be easily associated to the double-sided Feynman diagrams in Fig 1 in the main textOnce the state of the system is obtained the spectroscopy signals [SFS]

ω1ω2ω3ω4

e1e2e3e4(τ T t) synchronously detected

at φ = minusφ1 + φ2 + φ3 minus φ4 follow from the calculation of 〈A(τ + T + t)〉 = trAρω1ω2ω3ω4

e1e2e3e4(τ + T + t) with A =

sum

ν=eeprimef Γν |ν 〉〈 ν| Specifically

〈A(τ + T + t)〉 =minus Ceω4(microge middot e4)〈e

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ g〉minus Ceprime

ω4(microgeprime middot e4)〈eprime

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ g〉+ Ceprime

ω4(microef middot e4)(1 minus Γ)〈f

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ e〉+ Ce

ω4(microfeprime middot e4)(1minus Γ)〈f

∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ eprime〉

(B4)

After replacing the explicit functional form of the density matrix elements 〈ν∣

∣ρω1ω2ω3

e1e2e3(τ + T + t)

∣

∣ ν〉 in Eq (B4) andafter conveniently collecting terms Eq (B4) leads the 2D-FS signals in Eq (14) that are the main result of thisarticle

Appendix C Isotropic Averages

Before proceeding to the calculation of the isotropic average it is necessary to express the dipole transition op-erators in the molecular frame In doing so take as reference the transition dipole operator microeg = microegmz Hencemicroeprimeg = microeprimeg cos(θeprimeg)mz + microeprimeg sin(θeprimeg)mx microfe = microfe cos(θfe)mz + microfe sin(θfe)mx and microfeprime = microfeprime cos(θfeprime)mz +microfeprime sin(θfeprime)mx The angle between the different transition dipole moments is given by

tan(θνmicro) =|microeg times microνmicro|microeg middot microνmicro

(C1)

The isotropically averaged signals can then be written in the compact form

Ppq = M

pqχ

pq p q isin e eprime (C2)

with

Pee(T ) =

[

〈P eeeezzzz(0 T 0)〉iso 〈P eeeeprime

zzzz (0 T 0)〉iso 〈P eeeprimeezzzz (0 T 0)〉iso 〈P eeeprimeeprime

zzzz (0 T 0)〉iso]

Peprimeeprime(T ) =

[

〈P eprimeeprimeeezzzz (0 T 0)〉iso 〈P eprimeeprimeeeprime

zzzz (0 T 0)〉iso 〈P eprimeeprimeeprimeezzzz (0 T 0)〉iso 〈P eprimeeprimeeprimeeprime

zzzz (0 T 0)〉iso]

Peeprime (T ) =

[

〈P eprimeeeezzzz (0 T 0)〉iso 〈P eprimeeeeprime

zzzz (0 T 0)〉iso 〈P eprimeeeprimeezzzz (0 T 0)〉iso 〈P eprimeeeprimeeprime

zzzz (0 T 0)〉iso

〈P eeprimeeezzzz (0 T 0)〉iso 〈P eeprimeeeprime

zzzz (0 T 0)〉iso 〈P eeprimeeprimeezzzz (0 T 0)〉iso 〈P eeprimeeprimeeprime

zzzz (0 T 0)〉iso]

(C3)

and

χee(T ) =

[

χeeee(T ) χeprimeeprimeee(T )realχeeprimeee(T )imageχeeprimeee(T )]

χeprimeeprime(T ) =

[

χeeeprimeeprime(T ) χeprimeeprimeeprimeeprime(T )realχeeprimeeprimeeprime(T )imageχeeprimeeprimeeprime(T )]

χeeprime(T ) =

[

realχeeeeprime (T )realχeprimeeprimeeeprime(T )realχeeprimeeeprime(T )realχeprimeeeeprime (T )

imageχeeeeprime (T )imageχeprimeeprimeeeprime(T )imageχeeprimeeeprime(T )imageχeprimeeeeprime(T )]

(C4)

11

Mee11 = minus 2

15micro4eg M

ee12 = minus1

5micro4eg minus (1minus Γ)

1

15[cos(2θfeprime) + 2]micro2

feprimemicro2eg

Mee13 = M

ee14 = M

ee21 = M

ee22 = M

ee31 = M

ee32 = M

ee43 = M

ee44 = 0

Mee23 = M

ee33 = minus 1

15micro2eg (1 minus Γ) [3 cos(θfe) cos(θfeprime ) + sin(θfe) sin(θfeprime)] microfemicrofeprime + 3 cos(θeprimeg)microegmicroeprimeg

Mee24 = minusiMee

23 Mee34 = iMee

23

Mee41 = minus 1

15micro2eg

[cos(2θeprimeg) + 2]micro2eprimeg + (1minus Γ) [cos(2θfe) + 2]micro2

fe

Mee42 = minus 2

15micro2egmicro

2eprimeg [cos(2θeprimeg) + 2]

(C5)

Meprimeeprime

11 = minus2

5micro2egmicro

2eprimeg [cos(2θeprimeg) + 2] M

eprimeeprime

12 = minus1

5micro2eg

[cos(2θeprimeg) + 2]micro2eg + (1minus Γ) [cos(2θfeprime minus θeprimeg) + 2]micro2

feprime

Meprimeeprime

13 = Meprimeeprime

14 = Meprimeeprime

21 = Meprimeeprime

22 = Meprimeeprime

31 = Meprimeeprime

32 = Meprimeeprime

43 = Meprimeeprime

44 = 0

Meprimeeprime

23 = Meprimeeprime

33 = minus 1

15micro2eprimeg (1 minus Γ) [2 cos(θfe minus θfeprime) + cos(θfe + θfeprime minus 2θeprimeg)]microfemicrofeprime + 3 cos(θeprimeg)microegmicroeprimeg

Meprimeeprime

24 = minusiMeprimeeprime

23 Meprimeeprime

34 = iMee23

Meprimeeprime

41 = minus 1

15micro2eprimeg

3micro2eprimeg + (1minus Γ) [cos(2θfe minus θeprimeg) + 2]micro2

fe

Meprimeeprime

42 = minus2

5micro4eprimeg

(C6)

Meeprime

11 = minus2

5cos(θeprimeg)micro

3egmicroeprimeg M

eeprime

12 = minus 1

15microegmicroeprimeg

3 cos(θeprimeg)micro2eg + (1 minus Γ) [cos(2θfeprime minus θeprimeg) + 2 cos(θeprimeg)]micro

2feprime

Meeprime

13 = Meeprime

14 = Meeprime

17 = Meeprime

18 Meeprime

16 = iMee11 M

eeprime

17 = iMee12

Meeprime

21 = Meeprime

22 = Meeprime

23 = Meeprime

25 = Meeprime

26 = Meeprime

27 = 0

Meeprime

24 = M eprimeeprime

33 = minus 1

15microegmicroeprimeg (1minus Γ) [2 cos(θfe minus θfeprime minus θeprimeg) + 2 cos(θfe) cos(θfeprime minus θeprimeg)] microfemicrofeprime

Meeprime

28 = minusiMeprimeeprime

24

Meeprime

34 = iMee23 M

eeprime

31 = Meeprime

32 = Meeprime

34 = Meeprime

35 = Meeprime

36 = Meeprime

38 = 0 Meeprime

37 = iMee23

Meeprime

41 = minus 1

15microegmicroeprimeg

3 cos(θeprimeg)micro2eprimeg + (1minus Γ) [cos(2θfe minus θeprimeg) + 2 cos(θeprimeg)]micro

2fe

Meeprime

42 = minus2

5cos(θeprimeg)microegmicro

3eprimeg

Meeprime

43 = Meeprime

44 = Meeprime

47 = Meeprime

48 = 0 Meeprime

45 = iMee41 M

eeprime

46 = iMee42

Meeprime

51 = Meeprime

11 Meeprime

52 = Meeprime

12 Meeprime

53 = Meeprime

54 = Meeprime

57 = Meeprime

58 = 0 Meeprime

55 = minusiMee11 M

eeprime

56 = minusiMee12

Meeprime

61 = Meeprime

62 = Meeprime

64 = Meeprime

65 = M eeprime

66 = Meeprime

67 = Meeprime

68 = 0 Meeprime

63 = Meprimeeprime

24 Meeprime

67 = minusiM eprimeeprime

24

Meeprime

71 = Meeprime

72 = Meeprime

73 = Meeprime

75 = Meeprime

76 = Meeprime

77 = 0 Meeprime

78 = minusMeprimeeprime

37

Meeprime

81 = Meeprime

11 Meeprime

82 = Meeprime

42

Meeprime

83 = Meeprime

84 = Meeprime

87 = Meeprime

88 = 0 Meeprime

85 = minusMee45 M

eeprime

86 = minusMee86

(C7)

In defining the vector χ(T ) besides its properties in Eq (2)-(4) the following relation were instrumental χggee(T )minus1 = minusχeeee(T )minus χeprimeeprimeee(T ) χggeprimeeprime(T )minus 1 = minusχeeeprimeeprime(T )minus χeprimeeprimeeprimeeprime(T ) and χggeeprime (T ) = minusχeeeeprime (T )minus χeprimeeprimeeeprime(T )