Ultra-Fast Relaxation, Decoherence and Localization of Photoexcited Statesin π-Conjugated Polymers: A TEBD Study

Jonathan R. Mannouch,1, 2 William Barford,1 and Sarah Al-Assam31)Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford, OX1 3QZ,United Kingdom2)University College, University of Oxford, Oxford, OX1 4BH, United Kingdom3)Department of Physics, Clarendon Laboratory, University of Oxford, Oxford, OX1 3PU,United Kingdom

The exciton relaxation dynamics of photoexcited electronic states in poly(p-phenylenevinylene) (PPV) aretheoretically investigated within a coarse-grained model, in which both the exciton and nuclear degrees offreedom are treated quantum mechanically. The Frenkel-Holstein Hamiltonian is used to describe the strongexciton-phonon coupling present in the system, while external damping of the internal nuclear degrees offreedom are accounted for by a Lindblad master equation. Numerically, the dynamics are computed usingthe time evolving block decimation (TEBD) and quantum jump trajectory techniques. The values of themodel parameters physically relevant to polymer systems naturally lead to a separation of time scales, withthe ultra-fast dynamics corresponding to energy transfer from the exciton to the internal phonon modes (i.e.,the C-C bond oscillations), while the longer time dynamics correspond to damping of these phonon modesby the external dissipation. Associated with these time scales, we investigate the following processes thatare indicative of the system relaxing onto the emissive chromophores of the polymer: 1) Exciton-polaronformation occurs on an ultra-fast time scale, with the associated exciton-phonon correlations present withinhalf a vibrational time period of the C-C bond oscillations. 2) Exciton decoherence is driven by the decay in thevibrational overlaps associated with exciton-polaron formation, occurring on the same time scale. 3) Excitondensity localization is driven by the external dissipation, arising from ‘wavefunction collapse’ occurring as aresult of the system-environment interactions. Finally, we show how fluorescence anisotropy measurementscan be used to investigate the exciton decoherence process during the relaxation dynamics.

I. INTRODUCTION

Upon photoexcitation of a conjugated polymer witha pulse of electromagnetic radiation, the exciton stateformed is no longer stabilised by the ground state nu-clear geometry, leading to coupled exciton-phonon dy-namics as the system relaxes back to equilibrium. Thesedynamics have been investigated using a wide array oftime-resolved spectroscopic techniques for the polymerpoly(p-phenylenevinylene) (PPV), including fluorescencedepolarization,1–3 three-pulse photon-echo4–6 and coher-ent electronic two-dimensional spectroscopy,7 with mul-tiple time scales being identified. Based on experimentaland theoretical studies,2,8,9 the largest time scale seenin these experiments of >1 ps is widely accepted to cor-respond to Förster type exciton energy transfer betweenchromophores. The two shorter time scales of

2

the coupled exciton-phonon dynamics therefore requiresa full quantum mechanical treatment of the system, asdescribed in this paper.

For one dimensional strongly correlated systems, thedensity matrix renormalization group technique hasbeen successful in obtaining accurate numerical resultsfor properties associated with the ground state of thesystem.21,22 Using the same Hilbert space truncationscheme, the time evolving block decimation (TEBD)technique23–25 is an efficient and accurate method forcomputing the short time dynamics of such systems. Oneappealing feature of this method is that the associatedcomputational cost scales linearly with the number of‘sites’ in the system,23,25 allowing in principle the quan-tum dynamics for large systems to be simulated. Indeed,the technique has already been applied to obtain accu-rate results for the charge dynamics within the one di-mensional Holstein model.26,27

Exciton relaxation dynamics for photoexcited states ofPPV are modeled in this paper using the Frenkel-Holsteinmodel and TEBD method. The phonon modes to whichthe exciton is coupled are assumed to be those associ-ated with the high frequency C-C bond oscillations. Thelow frequency torsional modes are neglected (althoughtheir possible role in relaxation and decoherence is dis-cussed in Sec. V). We model environmental and confor-mational static disorder through the exciton on-site en-ergies and hopping integrals, which leads to the excitondensity associated with the low energy eigenstates of thismodel being spatially Anderson localized,28 thus defin-ing the emissive chromophores of the polymer. Externaldamping of the internal phonon modes is included withina Lindblad master equation approach, which allows thesystem to dissipate energy to the environment and thusrelax into the low energy eigenstates.

This paper is structured as follows. In Sec. II, we showhow a polymer chain can be represented by a coarse-grained model, generating a reduced dimensional Frenkelexciton Hilbert space in which to model the associateddynamics. In Sec. III, the Frenkel-Holstein Hamiltonianis introduced for this coarse-grained model, which de-scribes the important interactions associated with theexciton and nuclear degrees of freedom. In addition, theLindblad master equation is introduced along with thenumerical techniques used to solve the underlying equa-tions. We then present our results in Sec. IV, includinga discussion of the various processes associated with theshort time relaxation dynamics, such as exciton-polaronformation, exciton decoherence and exciton density lo-calization. We also make a connection between theseprocesses and fluorescence depolarization measurements.Finally, we conclude in Sec. V, with a particular emphasison comparing our results to experiment.

FIG. 1. The mapping of a polymer chain conformation to acoarse-grained linear site model. Each site corresponds to amoiety along the polymer chain, with the connection betweensites characterised by the torsional angle, φn. Also shown isthe dipole unit vector, sn, associated with site/moiety n.

II. POLYMER CONFORMATIONS ANDCOARSE-GRAINING

Polymers exist in many conformations, arising fromfluctuations in the torsional angles around the singlecarbon-carbon bonds, as well as defects in the polymerchains themselves. While in the solid state these torsionalangle fluctuations are quasi-static, in solution they giverise to dynamical disorder. However, as the time scalefor rotation around these single carbon-carbon bonds ismuch longer than the associated time scale for excitonrelaxation, the observables associated with the excitonrelaxation dynamics can be calculated by averaging overmany different static polymer chain conformations forboth phases.

Figure 1 illustrates how each polymer conformation ofPPV is generated. The polymer chain is built up from thestarting moiety, a phenylene unit, using the parametervalues given in Table I, along with the following rules:

• The torsional angles, φn, are taken as Gaussianrandom variables, with a mean 〈φ〉 and standarddeviation σφ.

• With equal probability, the torsional angles, φn,and the vinylene bond angles, θn, can be positiveor negative, corresponding to an anticlockwise orclockwise rotation respectively.

• For a pure polymer chain, all vinylene units takethe lowest energy trans geometry. However, trans-cis defects are introduced randomly along thechain, with probability xcis.

In general, polymers contain many thousands of atomsand therefore performing a full atomistic simulation ofexciton dynamics for each polymer conformation is un-feasible. Our model can be coarse-grained, as has beendone in previous work,12 by representing a polymer chain

3

as a linear array of sites, where each site corresponds toone of the moieties of the polymer. This mapping is il-lustrated in Fig. 1.

In conjugated polymer systems, such as PPV, the largeelectron-electron interactions lead to the electron-holepair of the exciton being tightly bound, typically occu-pying the same moiety of the chain.29 This is known asa Frenkel exciton. Thus in our coarse-grained model,we only need to consider how the center of mass of theFrenkel exciton propagates along the array of sites. Asingle exciton state is kept per site, corresponding to thelowest energy excitation of each moiety.

Parameter Value Parameter Value

〈φ〉 15◦ σφ 5◦

xcis 0.08 θ 60◦

TABLE I: Parameters used to generate the conformationsassociated with PPV polymer chains.

III. MODELS AND NUMERICAL TECHNIQUES

In this section we introduce the model Hamiltonians,as well as outlining the various numerical techniques usedto solve the associated exciton dynamics.

A. The Frenkel Model

In previous work, the initial exciton state of a PPVpolymer, generated after photoexcitation, has been de-scribed by the Frenkel Hamiltonian:30

ĤF =∑n

�nâ†nân +

∑n

Jn

(â†n+1ân + â

†nân+1

)(1)

where â†n (ân) is the exciton creation (destruction) oper-ator, which creates (destroys) a Frenkel exciton on moi-ety n of the polymer chain. For PPV, the odd and evensites, n, correspond to phenylene and vinylene moietiesrespectively. Within this Hamiltonian, the on-site energyis given by:

�n = E0 + (−1)n∆

2+ αn (2)

where E0 is the average moiety excitation energy, ∆ isthe difference in the excitation energy between pheny-lene and vinylene moieties and αn represents the diago-nal disorder, with standard deviation σα. This diagonaldisorder arises physically from density fluctuations in theenvironment around the polymer chain, due to the inho-mogeneity of the material.

Additionally, the nearest neighbor exciton hopping in-tegrals present in the Frenkel Hamiltonian are givenby:31,32

Jn = JDD + JSE cos2 φn (3)

where φn are the torsional angles between moieties alongthe polymer chain. From Eq. (3), we see that there aretwo contributions to the nearest neighbor hopping inte-gral. The first contribution arises from a through spacedipole-dipole interaction, which is incorporated throughthe term JDD. While in principle this interaction term isnon zero between all moieties in the polymer chain, forthis analysis we only keep the nearest neighbor interac-tions, which are the dominant terms. The second con-tribution to the hopping integral arises from a throughbond super exchange interaction, in which the systempasses through an intermediate charge transfer excitonstate. The strength of this interaction depends on thetorsional angle between the moieties (∝ cos2 φn, wherecosφn gives the overlap of the pz orbitals on each moi-ety). Thus torsional fluctuations in the polymer chaingive rise to disordered hopping integrals in the FrenkelHamiltonian.

The disorder present in the model leads to the FrenkelHamiltonian having two forms of eigenstates. The lowenergy states of the Hamiltonian are Anderson localized,non-overlapping and nodeless states, called local excitonground states (LEGSs).33,34 A LEGS is quantified by asigned-value parameter, α, defined as:

α =

∣∣∣∣∣∑n

|ψn|ψn

∣∣∣∣∣ (4)where ψn is the probability amplitude of the Frenkel ex-citon being on site n. In accord with previous work,we define a LEGS as satisfying α ≥ 0.95.33,34 The ex-citon density for three LEGSs corresponding to a par-ticular conformation of a 99 moiety PPV polymer chainare given by the dotted curves in Fig. 2. It is the widthof these states that define the size of the polymer’s ab-sorbing chromophores.35 In addition, the Frenkel Hamil-tonian has higher energy eigenstates that are delocalizedover several chromophores, called quasi-extended excitonstates (QEES).30 The exciton density for a QEES on thesame 99 moiety PPV polymer chain is given in Fig. 3.

B. The Frenkel-Holstein Model

Relaxation of a LEGS or QEES is physically inducedby exciton-nuclear coupling, which allows the nuclear ge-ometry of the polymer to distort so that it best stabilisesthe newly formed excited electronic state. Most vibra-tional normal modes in a polymer such as PPV eithercouple weakly to the exciton or have a low vibrationalfrequency, such that they do not strongly influence theexciton dynamics on the ultra-fast time scale of inter-est and therefore can be neglected in our model. As inprevious work, we thus consider the Frenkel exciton cou-pling to a single high frequency normal mode per moiety(the benzenoid-quinoid distortion in the phenylene unitand the C-C double bond stretch in the vinylene unit)

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FIG. 2. The three LEGSs (dotted lines) and three VRSs(solid lines) for one particular conformation of a PPV polymerchain made up of 99 moieties. The exciton center-of-massquantum number, j, for each state is also given. The VRSswere obtained using the density matrix renormalization grouptechnique, allowing a maximum of two phonons per site.

Moiety Number

0 20 40 60 80 100

Exc

iton

Den

sity

0

0.015

0.03

0.045

j = 7

FIG. 3. The associated exciton density of a QEES, withquantum number j = 7, for one particular conformation of aPPV polymer chain made up of 99 moieties.

modeled using the Frenkel-Holstein Hamiltonian:10,11

ĤFH = ĤF−Ah̄ω√

2

∑n

â†nân

(b̂†n + b̂n

)+h̄ω

∑n

(b̂†nb̂n +

12

)(5)

where A is the dimensionless exciton-nuclear couplingparameter, which determines the strength of the exciton-phonon coupling, and h̄ω is the phonon energy associatedwith the vibrational normal modes. Within the Frenkel-Holstein model, the vibrational normal mode on moi-ety/site n is represented by a harmonic oscillator, where

b̂†n (b̂n) is the associated dimensionless harmonic oscilla-tor creation (destruction) operator.

One effect of the exciton-nuclear coupling term in thisHamiltonian is to further localize the exciton density ofthe low energy LEGSs, to create so-called vibrationallyrelaxed states (VRSs).36 These VRSs are the low energyeigenstates of the Frenkel-Holstein Hamiltonian,37 withtheir associated exciton densities given by the solid curvesin Fig. 2 for a 99 moiety PPV polymer chain. Com-paring these VRSs with their associated LEGSs (givenby the dotted curves) shows that the degree of self-localization arising from coupling to the high frequencyvibrational modes is small, leading to the spatial extentof the VRSs still being largely determined by the disor-der in the Frenkel Hamiltonian.36,37 It is the width ofthese VRSs that define the size of the polymer’s emissivechromophores.12

The values for the Frenkel-Holstein Hamiltonian pa-rameters that we used to model PPV12 are given in Ta-ble II.

Parameter Value Parameter Value

E0 9.24 eV JDD −1.35 eV

∆ 3.20 eV h̄ω 0.2 eV

σα 65 meV A 4.00

JSE −1.96 eV γ̃ 0.033

TABLE II: Parameters used in the Frenkel-HolsteinHamiltonian.

C. The Lindblad Master Equation

To describe the time evolution of a Frenkel exciton(initially in a high energy QEES or LEGS) relaxing tothe low energy VRSs, the model must allow the systemto lose excess energy to the environment. Dissipationof energy arising from system-environment coupling iscommonly accounted for by describing the time evolutionof the quantum system by a Lindblad master equation:38

∂ρ̂

∂t̃= − i

h̄ω

[Ĥ, ρ̂

]− γ̃

2

∑n

(b̂†nb̂nρ̂+ ρ̂b̂

†nb̂n − 2b̂nρ̂b̂†n

)(6)

where t̃ = tω is the dimensionless time. Such an equa-tion describes the time dependence of the system densitymatrix, ρ̂, from which any observable associated with thesystem of interest can be obtained. From Eq. (6), we seethat the time dependence of the system density matrixdepends on two terms. The first term on the right ofEq. (6) corresponds to the evolution of the system den-

sity matrix under the Hamiltonian, Ĥ, given by:

Ĥ = ĤFH +γ̃h̄ω

4

∑n

(Q̂nP̂n + P̂nQ̂n

)(7)

where Q̂n =1√2(b̂†n + b̂n) is the dimensionless displace-

ment operator and P̂n =i√2(b̂†n − b̂n) is the dimension-

5

less momentum operator, for the oscillator on site n, andthe last term in Eq. (7) accounts for the damping cor-rection to the harmonic oscillator frequency induced bythe external dissipation.39 The second term in Eq. (6) ac-counts for energy dissipation of the system induced by thesystem-environment coupling. The effect of the system-environment coupling is controlled by the dimensionlessdissipation parameter γ̃ = γω , as well as the Lindblad op-

erator for site n, b̂n, which for the system considered inthis paper becomes the dimensionless harmonic oscillatordestruction operator.

While the Lindblad master equation only approx-imately describes dissipation effects due to system-environment coupling, the perturbative nature of itsderivation means that we expect the master equation todescribe the dynamics of systems in which the system-environment coupling is weak (i.e., when γ̃ is small).Such a situation is indeed physically relevant to the exci-ton relaxation dynamics in polymer chains, in which theexciton-phonon coupling strength is much larger that anycoupling terms to the environment.

It is not necessarily obvious why the harmonic oscilla-tor destruction operator is a good choice for the Lindbladoperator in our master equation in Eq. (6). To show thatit is, we first note that the expectation value of some op-erator corresponding to a system observable, Ô, can becalculated from the system density matrix as follows:

〈Ô〉 = Tr[ρ̂Ô]

(8)

where Tr[· · · ] corresponds to taking the trace of the quan-tity inside the brackets. Using Eq. (6), we can then derivethe following equations of motion for the dimensionlessnuclear displacement and momentum:

d〈Q̂n〉dt̃

= 〈P̂n〉

d〈P̂n〉dt̃

= A〈â†nân〉 − 〈Q̂n〉 − γ̃〈P̂n〉(9)

which have the form of Newton’s equations of motion fora damped harmonic oscillator38,39 and are the identicalequations of motion used previously for modeling excitonrelaxation dynamics for classical nuclei.12

D. Optical Intensity

In an experiment, the initial exciton state from whichthe relaxation dynamics proceed is formed by photoexci-tation (typically with a broadband pulse) from the elec-tronic ground state. On application of this pulse, thenuclear degrees of freedom remain static (i.e., the har-monic oscillators remain in their ground state), while theexciton is created in a state that corresponds to a linearcombination of the eigenstates of the Frenkel Hamilto-nian. The probability that an eigenstate of the Frenkel

Hamiltonian with energy E contributes to this initial ex-citon state is given by the spectral function:

I (E) =

〈∑α

fαδ (E − Eα)

〉Disorder

(10)

where Eα is the energy eigenvalue associated with eigen-state |ψα〉 of the Frenkel Hamiltonian. In addition, fαis the associated oscillator strength for this eigenstate,given by:

fα =

(2meEα

3e2h̄2

) ∑s=x,y,z

|〈ψα|µ̂s|0〉|2 (11)

where |0〉 corresponds to the electronic ground state ofthe system. On calculating the spectral function usingEq. (10), we average our result over the various instancesof the disorder in the Frenkel Hamiltonian, as well asover all possible conformations of the polymer chain. Fora particular conformation, the transition dipole momentoperator, µ̂, takes the form:

µ̂ = µ0∑n

sn(â†n + ân

)(12)

where µ0 is the magnitude of the transition dipole mo-ment for a Frenkel exciton localized on a single moietyof the polymer and â†n (ân) is the exciton creation (de-struction) operator for site/moiety n. Additionally, snis a unit vector that points along the polymer axis atmoiety n, which is illustrated in Fig. 1.

Figure 4 shows the calculated spectral function forPPV, using the parameter values given in Tables I and II.In this work, we chose to perform our relaxation dynam-ics for initial Frenkel exciton energies of 2.68 eV and2.88 eV, shown by the dotted lines in Fig. 4. Both ofthese Frenkel exciton energies have appreciable forma-tion probabilities and also allow us to investigate the dif-fering relaxation dynamics of LEGSs, low energy QEESsand high energy QEESs.

E. Numerical Techniques

Section III C outlines the necessary theoretical frame-work for modeling the exciton relaxation dynamics. How-ever, evolving the system density matrix numericallyusing the Lindblad master equation given in Eq. (6)is not straight forward for the following reasons. TheFrenkel-Holstein model contains exciton-phonon interac-tions, which lead to its associated Hilbert space growingexponentially with the number of sites/moieties in thepolymer chain. Thus for chains containing a physicallyrealistic number of moieties/sites, this Hilbert space istoo large to manipulate on a computer. This issue iscompounded further, as the size of the system densitymatrix is the Hilbert space squared, meaning the use ofstandard differential equation solvers to compute the sys-tem density matrix from Eq. (6) is intractable. In this

6

FIG. 4. The theoretically obtained spectral function, I(E),for PPV polymers, that corresponds to the probability thatFrenkel exciton states with energy E are produced by a broad-band photoexcitation pulse. The low energy part of the spec-trum is dominated by the LEGSs, whose spatial extent definethe size of the polymers absorbing chromophores. The dot-ted lines at 2.68 eV and 2.88 eV illustrate the initial Frenkelexciton energies from which the exciton relaxation dynamicsare simulated.

work, these issues are overcome by using the time evolv-ing block decimation (TEBD)24,25,40 and the quantumjump trajectory methods,41–44 respectively. The detailsof these techniques are not discussed here, but since theseare unfamiliar to the chemical physics community, anoverview of both can be found in the Appendices.

Before a discussion of our results, we briefly outlinethe basic constituents of the quantum jump trajectorymethod, which will prove useful in interpreting the roleof the external dissipation in our calculated dynamics.The quantum jump trajectory method obtains systemobservables associated with the numerical solution of aLindblad master equation, such as Eq. (6), by averagingover so-called quantum trajectories. A single trajectoryis represented by a state in the system Hilbert space, withits time evolution determined as follows. Typically, thetrajectory evolves under the effective Hamiltonian:41

Ĥeff = Ĥ −iγ̃

2

∑n

b̂†nb̂n (13)

where Ĥ is the system Hamiltonian given in Eq. (7).However, at probabilistically determined times, the tra-jectory undergoes a ‘quantum jump’, which correspondsto application of one of the Lindblad operators onto thestate. Averaging over a large number of these stochasti-cally determined trajectories is known to accurately re-produce the time-dependent observables associated withthe Lindblad master equation.41

t/fs

0 10 20 30 40 50 60

Ene

rgy

/ eV

2.4

2.5

2.6

2.7

2.8

2.9

VRSsE = 2:88 eVE = 2:68 eV

FIG. 5. The time dependence of the total system energy. Theblue curve corresponds to an initial Frenkel exciton energy of2.88 eV, while the red curve corresponds to an initial Frenkelexciton energy of 2.68 eV. The black dashed lines show thecorresponding system energy for the VRSs of the polymerchain for different instances of the disorder.

IV. RESULTS

The dynamics must fulfil certain criteria if the timeevolution determined by our theoretical methodology isphysically correct. One such requirement is that the totalenergy of the system must decrease during the dynamics,approaching the energies of the VRSs, which define theemissive chromophores of the polymer. We indeed findthat the system energy does decrease in our dynamicsmodel, as shown by the time evolution of the total en-ergy of the system for two different initial Frenkel excitonenergies given in Figure 5. This confirms that the formof the external dissipation in our model is having the de-sired effect. Also shown in this figure are the energiesassociated with the VRSs, for various instances of thedisorder, given by the dotted curves. The fact that thetotal system energy at t = 60 fs is greater than the energyof these VRSs for both initial Frenkel exciton energies,as well as dE/dt being non zero at this time, suggeststhat the dynamics are far from equilibrium. This is notsurprising, since the value of the exponential dissipationtime scale, γ−1 ∼ 100 fs, means that the system willonly reach equilibrium at times much longer than com-putationally practical simulations of ∼100 fs.

We observe a separation of time scales within our com-puted dynamics. This is illustrated in Fig. 6, which showsthe time-dependent phonon energy for two different ini-tial Frenkel exciton energies, with and without externaldissipation. For very short times, the evolution of thesystem observables depend on the Frenkel-Holstein pa-rameters and are largely independent of the external dis-sipation, γ̃. From Fig. 6, we see that the phonon en-ergy initially increases during the dynamics, suggesting

7

FIG. 6. The time dependence of the phonon energy. Theblue curves correspond to an initial Frenkel exciton energy of2.88 eV, while the red curves correspond to an initial Frenkelexciton energy of 2.68 eV. In addition, the dashed curves cor-respond to the relaxation dynamics with no energy dissipationto the environment, while the solid curves correspond to thedynamics with γ̃ = 0.033.

that this ultra-fast time scale is characterised by energytransfer from the exciton to the internal nuclear degreesof freedom (i.e., the phonon degrees of freedom act as aheat bath for the exciton). Eventually, the energy as-sociated with the nuclear degrees of freedom saturates,with the time evolution of the observables now dependenton the longer external dissipation time scale. Figure 6shows that for γ̃ 6= 0, the phonon energy decreases atlong times, suggesting that the long time scale dynam-ics are characterised by dissipation of the phonon energyto the environment (i.e., the environment acts as a heatbath for the internal nuclear degrees of freedom).

Even though the system is far from equilibrium for thetimes considered in our relaxation dynamics simulation,we should expect the time evolution of observables tostill exhibit features associated with the steady states.Indeed, if the system does relax to the low energy VRSsas physically expected, processes such as exciton-polaronformation, exciton decoherence and exciton density lo-calization should be present in the short time dynamics.We now consider each of these processes.

A. Exciton-Polaron Formation

In systems with strong electron-phonon interactions,the low energy excitations of the system are described bypolarons, which are quasiparticles consisting of an elec-tron or exciton ‘dressed’ or ‘self-trapped’ by local dis-placements of the nuclei.45 (To distinguish dressed elec-trons and excitons, we use the term ‘exciton-polaron’ inthis paper for the dressed exciton.) The effective size

FIG. 7. The time dependence of the exciton-phonon corre-lation function, Cex-phn , for an initial high energy QEES givenin Fig. 3.

of the exciton-polaron can be described by the exciton-phonon correlation function:46

Cex-phn =1

A

∑m

〈â†mâmQ̂m+n〉 (14)

where the factor 1/A normalizes the function foreigenstates of the Frenkel-Holstein Hamiltonian (i.e.,∑n C

ex-phn = 1).

47 Physically, this correlation functioncorresponds to the average nuclear displacement n sitesaway from the exciton. From previous work, it is knownthat for the VRSs of the Frenkel-Holstein model, theexciton-phonon correlation function has an exponentialform, with a correlation length that is a function of theparameters h̄ω, A and 〈J〉.37 If the system does evolveto the VRSs, then we would expect the exciton-phononcorrelation function to acquire this exponential behaviorduring the exciton relaxation dynamics.

Figure 7 shows the time evolution of the exciton-phonon correlation function for an initial high energyQEES given in Fig. 3. This correlation function acquiresan exponential form during the dynamics, confirmingthat exciton-polaron formation is occurring within ourmodel. Exciton-polaron formation occurs on an ultra-fast time scale (∼10 fs), with the dynamics being largelyindependent of the external dissipation. Indeed, we findthat the time scale associated with exciton-polaron for-mation solely depends on the parameter h̄ω, with the lo-cal exciton-phonon correlations present within half a vi-brational time period. In addition, persisting oscillationsoccur in the exciton-phonon correlation function with atime period (∼20 fs) that matches the vibrational timeperiod of the harmonic oscillators in the model. Theseoscillations thus arise from the time evolution of the nu-clear displacements Q̂n, which physically correspond tothe C-C bond oscillations.

8

FIG. 8. The exciton-phonon correlation function, Cex-phn ,at t = 60 fs for two initial photoexcitation energies (2.68 eVand 2.88 eV) as well as several instances of the disorder inthe Frenkel-Holstein Hamiltonian (given by the solid curves).Also shown is the average exciton-phonon correlation functionfor the VRSs, given by the dashed curve.

We now consider whether the correlation length as-sociated with the dynamical exciton-phonon correlationfunction given in Fig. 7 is the same as for the VRSs. Thesolid curves in Fig. 8, which correspond to the exciton-phonon correlation functions at t = 60 fs for two initialFrenkel exciton energies (2.68 eV and 2.88 eV), as well asseveral instances of the disorder, all coincide. This is tobe expected, as the parameters h̄ω, A and 〈J〉 have thesame values for all these curves, giving rise to an identicalexciton-polaron formation time scale and exciton-phononcorrelation length. Also shown in Fig. 8 is the averageexciton-phonon correlation function associated with theVRSs, given by the dashed curve. The form of the dy-namical exciton-phonon correlation functions at t = 60 fsshows good agreement with the same function for theVRSs, which is consistent with the system evolving tothese states during the dynamics. However the agree-ment is not perfect, with deviations arising due to thepersistent oscillations in the dynamical correlation func-tion, which are symptomatic of the time evolution havingnot yet reached the steady state.

B. Exciton Decoherence

An exciton confined to a one dimensional polymerchain in general can have long range quantum coher-ences between the moieties, which physically give rise tointerference effects in the calculated observables. Whilethe range of these coherences are often limited by thepresence of disorder in the system, for QEESs the exci-ton coherences can still persist over a distance of severalchromophores. If the exciton localizes onto a single chro-

mophore during the relaxation dynamics, then these longrange coherences will decay.

One way to quantify the magnitude of the exciton co-herences is from the off-diagonal elements of the excitonreduced density matrix, ρ̂ex:

ρ̂ex =∑v

〈v|ρ̂|v〉 (15)

which is obtained by taking the trace of the system den-sity matrix over the nuclear degrees of freedom (charac-terised by the quantum number v). The following excitoncoherence correlation function can then be defined, whichgives the average magnitude of the exciton coherences be-tween moieties/sites a distance n apart:48,49

Ccohn =∑m

|〈m|ρ̂ex|m+ n〉| (16)

where |m〉 is the ket corresponding to the exciton onsite m.

Figure 9 shows the time dependence of the excitoncoherence correlation function for an initial high energyQEES given in Fig. 3. The correlation function rapidlylocalizes, showing that within our model, decoherence ofthe exciton occurs on an ultra-fast time scale. This timescale is more evident from the time dependence of theexciton coherence number, Ncoh:

Ncoh =∑n

Ccohn (17)

which corresponds to the average number of moieties overwhich exciton coherences persist and is given in the in-set of Fig. 9. Indeed, we find that the coherence num-ber reaches its equilibrium value within ∼10 fs. As forexciton-polaron formation, the short time scale associ-ated with this decoherence leads to the associated dy-namics being largely independent of the external dissi-pation and depending solely on the Frenkel-Holstein pa-rameters.

The exciton decoherence mechanism present withinour model can be elucidated by considering the followingstate representation for the Frenkel-Holstein model:

|ψ〉 =∑n

ψn |n〉 |Vn〉 (18)

Within this representation, ψn corresponds to the prob-ability amplitude for the Frenkel exciton residing onsite/moitey n, given by |n〉, while the ket |Vn〉 corre-sponds to the state of the L harmonic oscillators associ-ated with the exciton on site n. Using this state represen-tation, the exciton coherence correlation function takesthe form:

Ccohn =∑m

∣∣ψmψ∗m+n 〈Vm+n|Vm〉∣∣ (19)This expression contains two terms. The first term,ψmψ

∗m+n, corresponds to the exciton wavefunction over-

lap between the two sites/moieties. As shown in

9

FIG. 9. The time dependence of the exciton coherence cor-relation function, Ccohn , for an initial high energy QEES givenin Fig. 3. In addition, the time dependence of the associatedcoherence number, Ncoh, is given in the inset figure.

Sec. IV C, this term remains essentially stationary overthe time scale of exciton decoherence. The second term,〈Vm+n|Vm〉, corresponds to the overlap of the vibrationalstates associated with the exciton being on sites/moietiesm + n and m. In Sec. IV A, we saw that during theexciton relaxation dynamics, exciton-polaron formationoccurs on an ultra-fast time scale, leading to the nucleardisplacements of the polymer becoming locally correlatedto the exciton. This means that during the dynamics,the vibrational state |Vm〉 will only have nuclear displace-ments spatially close to moiety/site m (where the excitonresides), thus causing the vibrational overlap 〈Vm+n|Vm〉to decrease with increasing site separation n. It is thislocal nature of the vibrational overlap that accounts forthe local nature of the exciton coherence correlation func-tion in Fig. 9. Exciton-polaron formation is thereforethe mechanism by which the exciton decoheres, which isconsistent with decoherence mechanisms found in otherelectron-nuclear coupled systems.16–18

Knowledge of the exciton decoherence mechanism nowgives insight into the dependence of the associated timescale on the Frenkel-Holstein parameters. Based on thismechanism, we would expect the exciton decoherencetime scale to be identical to that for exciton-polaron for-mation, depending solely on h̄ω and with decoherenceoccurring within half a vibrational time period. Whilethis is consistent with our results, we find that the exci-ton decoherence time scale additionally depends on theexciton-phonon coupling strength, A, with the time scaledecreasing with increasing A. This arises, because in-creasing A leads to an increase in the nuclear displace-ments associated with the exciton-polaron, which resultsin a more rapid decay in the vibrational overlaps thatlead to decoherence.

FIG. 10. The exciton density at t = 100 fs, given by the solidcurve, for an initial QEES given in Fig. 3. Also present arethe scaled exciton densities for the three VRSs of the polymerchain, given by the dotted curves.

C. Exciton Density Localization

In Sec. IV B, we saw that rapid exciton decoherenceoccurs during the relaxation dynamics, driven by the de-cay in the overlap of the vibrational states associatedwith exciton-polaron formation. As exciton decoherencearises from the vibrational configuration of the system,the exciton density can in principle be delocalized alongthe polymer chain, even if the exciton coherences betweendifferent sites/moieties are short ranged. Therefore, if thesystem does relax to the low energy VRSs, the excitondensity must also localize during the time evolution.

In Fig. 10, the solid curve corresponds to the excitondensity at time t = 100 fs for an initial high energy QEESgiven in Fig. 3. Also shown are the scaled exciton densi-ties for the three VRSs of the polymer chain, given by thedotted curves. Three of the peaks in the exciton densityat t = 100 fs match those corresponding to the VRSs,suggesting that the system is relaxing into these low en-ergy states during the time evolution. The presence ofadditional peaks in the time evolved exciton density areprobably a result of the system having not reached thesteady state at t = 100 fs, but could also signify thatthe steady state solutions of our Lindblad master equa-tion do not correspond to the VRSs. As we are primarilyinterested in understanding the short time dynamics inthis paper, we will pursue this potential issue further ina subsequent paper, where the steady state solutions ofthe Lindblad master equation will be investigated.

While the time evolved exciton density contains peakscorresponding to the VRSs of the polymer chain, the solidcurve in Fig. 10 seems to show that the exciton densityremains quasi-delocalized during the dynamics. How-ever, as the system is described by a mixed state density

10

matrix, the associated observables physically correspondto an ensemble average over many different environmentconfigurations (or alternatively, over many quantum tra-jectories). This means that it is not a priori obviouswhether the non-local nature of the ensemble averagedexciton density is symptomatic of an actual absence ofexciton density localization in our model or whether it issimply a consequence of ensemble averaging over excitondensity that has localized onto different chromophores.To distinguish between these two possibilities requires acorrelation function that measures the spatial extent ofthe exciton for a single environment configuration, suchas:50,51

C locn =

∑m |〈m, 0|ρ̂|m+ n, 0〉|∑m |〈m, 0|ρ̂|m, 0〉|

(20)

where the scaling factor is chosen so that C loc0 = 1. In thisdefinition, the ket |m, 0〉 corresponds to the exciton beingon site/moiety m, while all the L harmonic oscillators arein their ground state, signified by the ‘0’ index.

The physical interpretation of this exciton localizationcorrelation function can be understood by consideringthe state representation for the Frenkel-Holstein model,given in Eq. (18). Using this state representation, thecorrelation function takes the form:

C locn =

∑m

∣∣ψmψ∗m+n 〈Vm+n|0〉 〈0|Vm〉∣∣∑m |ψm|

2 |〈Vm|0〉|2(21)

where ψm is the probability amplitude for the excitonbeing on moiety/site m, |Vm〉 corresponds to the stateof the nuclear degrees of freedom when the exciton re-sides on site m and |0〉 corresponds to the ground stateof the L harmonic oscillators in the system. To a goodapproximation, the quantity 〈Vm|0〉 will be independentof the exciton site index m. This is because the spatialdistribution of the oscillator displacements around theexciton depend on the exciton-phonon correlation length,which will be largely independent of the exciton site in-dex. Applying this approximation to Eq. (21) leads tothe following simplified form for the exciton localizationcorrelation function:

C locn ≈∑m

∣∣ψmψ∗m+n∣∣ (22)This expression shows that the correlation function canbe regarded as giving the average magnitude of the ex-citon wavefunction overlap between moieties/sites a dis-tance n apart and therefore gives a measure of the spatialextent of the exciton on a polymer chain.

Figure 11 shows the time dependence of the exciton lo-calization correlation function for an initial high energyQEES given in Fig. 3. The main figure corresponds to thetime evolution with the standard external dissipation pa-rameter of γ̃ = 0.033, while the upper inset correspondsto the time evolution without external dissipation to theenvironment. When external dissipation is present, this

FIG. 11. The time dependence of the exciton localizationcorrelation function, C locn , for an initial high energy QEESgiven in Fig. 3. The main figure corresponds to the timeevolution with γ̃ = 0.033, while the upper inset figure corre-sponds to the time evolution without external dissipation. Inaddition, the time dependence of the associated exciton lo-calization number, Nloc, with γ̃ = 0.033 is given in the lowerinset figure.

correlation function does localize, confirming that exci-ton density localization is present within our model ofthe relaxation dynamics. However, the time scale asso-ciated with this process appears substantially differentfrom the ultra-fast exciton-polaron formation and exci-ton decoherence time scales studied previously. Indeed,the exciton localization correlation function remains es-sentially static for initial times on the order of the ultra-fast time scale (∼10 fs) with the function only startingto localize at longer times. This time scale is more evi-dent from the time dependence of the exciton localizationnumber, Nloc:

51

Nloc =∑n

C locn (23)

which corresponds to the average number of moieties overwhich the exciton wavefunction overlap remains non zeroand is given in the lower inset of Fig. 11. The excitonlocalization number continues to decrease for t > 60 fs,illustrating the much longer time scale associated withthis process. This suggests that the exciton density lo-calization is driven by the external dissipation to the en-vironment. Further confirmation of this is given in theupper inset of Fig. 11, which shows that without externaldissipation, the correlation function remains delocalizedduring the time evolution.

The mechanism by which external dissipation drivesthis exciton density localization can be understood byinvestigating the time evolution of individual trajecto-ries within the quantum jump trajectory method. Thequantum jump trajectory method is a way of simulat-

11

FIG. 12. The time dependence of the exciton density fora single trajectory of the quantum jump trajectory method.The discontinuity in the density at ca. 20 fs is a ‘quantumjump’ caused by the stochastic application of a Lindblad jumpoperator. The dynamics were performed for an initial highenergy QEES given in Fig. 3, with γ̃ = 0.033.

ing the time evolution described by a Lindblad masterequation and is outlined briefly in Sec. III E, with a moredetailed summary given in Appendix A. Figure 12 showsthe time dependence of the exciton density for a typicaltrajectory associated with this technique, where the sys-tem is initially in a high energy QEES given in Fig. 3.For t < 20 fs, the dynamics correspond to time evolutionunder the effective Hamiltonian, Ĥeff given in Eq. (13),where the figure shows that during this evolution, theexciton density remains quasi-delocalized along the poly-mer chain. The discontinuity in the exciton density att ∼ 20 fs corresponds to a ‘quantum jump’, in which oneof the harmonic oscillator destruction operators, b̂n, israndomly applied to the system. The effect of this ‘quan-tum jump’ process is to cause an immediate localizationof the exciton density, as seen in Fig. 12.52 This arises be-cause only states of the harmonic oscillator on site n witha non zero displacement remain after application of the

operator, b̂n. As these nuclear displacements are locallycorrelated to the exciton through exciton-polaron forma-

tion, application of b̂n therefore leads to a state with theexciton localized around site n. Physically this processcorresponds to ‘wavefunction collapse’, where local en-vironment interactions with the internal phonon degreesof freedom act as a quantum measurement. Hence, it isthe position of these local environment interactions thatdetermines the particular chromophore that the excitondensity relaxes onto during the time evolution.

D. LEGSs Dynamics

In the preceding analysis, the main focus has been onunderstanding the short time exciton relaxation dynam-ics of initial QEESs. However, the features of the dy-namics, as well as their associated time scales, are largelyidentical for initial LEGSs, with a few important differ-ences. For QEESs, we saw that the system evolves tobecome a mixed state combination of the various VRSson the polymer chain, as seen in the time evolved excitondensity. In contrast, we find that an initial LEGS adia-batically relaxes almost entirely onto a single VRS, withthe exciton density remaining on the same chromophorethroughout the time evolution. This finding is in agree-ment with previous work on the relaxation dynamics ofLEGSs.12,53 Finally, as the spatial extent of LEGSs andVRSs are very similar (see Fig. 2) the extent of excitondecoherence and exciton density localization during thetime evolution is much less pronounced compared to thatfor an initial QEES, resulting in a smaller time depen-dence of the associated correlation functions.

E. Time Resolved Fluorescence Anisotropy

So far, we have outlined many features present in ourmodel of the exciton relaxation dynamics in polymer sys-tems, such as exciton-polaron formation, exciton deco-herence and exciton density localization. We now con-sider whether such features can be seen in experimentalobservables used to study the dynamics.

One such experimental technique is the measure-ment of the time-dependent decay in the fluorescenceanisotropy after photoexcitation. It is known experimen-tally that as the exciton relaxes, the polarization axis as-sociated with the fluorescence rotates from that of theincident radiation, caused by a rotation of the transitiondipole moment of the polymer. This can be quantifiedusing the fluorescence anisotropy, r:54

r =I‖ − I⊥I‖ + 2I⊥

(24)

where I‖ and I⊥ are the intensities of the fluorescenceradiation polarised parallel and perpendicular to the in-cident radiation, respectively.

For an arbitrary state of a quantum system, |ψ〉, thefluorescence intensity polarised along the x axis is relatedto the x component of the transition dipole operator, µ̂x,by:

Ix ∝∑v

|〈ψ|µ̂x|0, v〉|2 (25)

where |0, v〉 corresponds to the system in the ground elec-tronic state, with the nuclear degrees of freedom in thestate characterised by the quantum number v. In princi-ple, the expression for Ix also has an energy dependent

12

FIG. 13. The time dependence of the fluorescence anisotropy,〈r (t)〉, for two initial Frenkel exciton energies. The red curvecorresponds to an initial Frenkel exciton energy of 2.68 eV,while the blue curve corresponds to an initial Frenkel excitonenergy of 2.88 eV

term. However, as long as the variance of the energy as-sociated with the system remains small during the timeevolution, this term just becomes a multiplicative con-stant that can be neglected in Eq. (25). The expressiongiven in Eq. (25) can be generalised for a mixed statedensity matrix, ρ̂, as follows:55

Ix ∝∑v

〈0, v|µ̂xρ̂µ̂x|0, v〉

∝∑m,n

sm,xsn,x 〈m|ρ̂ex|n〉(26)

where the final equation is obtained by using the expres-sion for the transition dipole moment given in Eq. (12).In this equation, ρ̂ex corresponds to the exciton reduceddensity matrix, defined in Eq. (15), while |n〉 is the ketcorresponding to the exciton residing on site n. In ad-dition, sn,x is the x component of the unit vector thatpoints along the polymer axis at site n, illustrated inFig. 1. Inserting the required intensity components,given in Eq. (26), into Eq. (24) allows the fluorescenceanisotropy to be obtained for a specific polymer confor-mation. The fluorescence anisotropy can then be aver-aged over several different polymer conformations andinitial exciton states using the following expression:54

〈r (t)〉 = 0.4×∑i Ii (t) ri (t)∑

i Ii (t)(27)

where Ii (t) is the total fluorescence intensity and ri (t) isthe fluorescence anisotropy, both associated with confor-mation i at time t. The factor of 0.4 is included on theassumption that the polymers are oriented uniformly inthe bulk material.54

Figure 13 shows the calculated time dependence of thefluorescence anisotropy for two different initial Frenkelexciton energies. The mechanism for the decay of thefluorescence anisotropy can be understood by compar-ing the expression for the components of the fluorescenceintensity given in Eq. (26) with the expression for theexciton coherence correlation function given in Eq. (16).While the fluorescence intensity components cannot bedirectly written in terms of the exciton coherence cor-relation function, since both quantities depend on theelements of the exciton reduced density matrix, the timedependence of the two quantities are related. Indeed, wefind numerically that the time dependence of the fluores-cence intensity components are dominated by the decayof the off diagonal elements of the exciton reduced den-sity matrix, which also give rise to the localization ofthe exciton coherence correlation function. The decay inthe fluorescence anisotropy can therefore be attributedto the decoherence of the exciton during the relaxationdynamics, induced by exciton-polaron formation. This isalso consistent with the observations that the decay ofthe fluorescence anisotropy is independent of γ̃ and alsooccurs on a time scale equal to the exciton decoherencetime scale from Fig. 9.

The difference in the magnitude of the fluorescenceanisotropy decay for the two initial Frenkel exciton ener-gies can now be explained in terms of this exciton deco-herence mechanism. For the lower initial Frenkel excitonenergy of 2.68 eV, Fig. 4 shows that a large fraction of theinitial Frenkel exciton states are LEGSs. As the excitonis already localized onto a single chromophore in a LEGS,the extent of exciton decoherence during the relaxationdynamics is much reduced for an initial LEGS comparedto an initial QEES. This therefore explains why the mag-nitude of the fluorescence anisotropy decay is smaller forthe lower initial Frenkel exciton energy in Fig. 13, con-sistent with experimental observations.1

V. DISCUSSION AND CONCLUSIONS

In this paper, we have introduced a model for excitonrelaxation dynamics in polymer systems. This model isbased on describing the system in terms of a parameter-ized Frenkel-Holstein Hamiltonian, which explicitly in-cludes exciton-phonon coupling. While this Hamiltonianhas been previously used to describe exciton relaxationdynamics in polymer systems,12 our approach differs aswe treat the nuclear degrees of freedom quantum mechan-ically, with external dissipation of these modes includedin a Lindblad master equation formalism. Our dynam-ics simulations are then performed numerically using thequantum jump trajectory and time evolving block deci-mation techniques.

We find that our initial results for the time evolutionare physically sensible, with the total system energy de-creasing as a function of time, confirming that the dis-sipation in our model is having the desired effect. The

13

values of the parameters relevant to polymer systems nat-urally lead to a separation of time scales within the dy-namics. The ultra-fast time scale depends on the Frenkel-Holstein parameters and physically corresponds to en-ergy transfer between the exciton and nuclear degreesof freedom, while the longer time dynamics are drivenby the external dissipation parameter, γ̃, correspondingto damping of these phonon modes by the environment.These time scales are also found to be relevant to severalfeatures of our dynamics model, such as exciton-polaronformation, exciton decoherence and exciton density lo-calization, which are symptomatic of the system relaxingto the so-called vibrational relaxed states (VRSs). Ingeneral we find:

• Exciton-polaron formation occurs within half avibrational time period of the internal phononmodes (i.e., the C-C bond oscillations), namelywithin 10 fs. We also showed that the time-evolved exciton-phonon correlation function is ingood agreement with that for the VRSs, whichis consistent with the dynamics relaxing to thesestates.

• Exciton decoherence is driven by the decay inthe vibrational overlaps associated with exciton-polaron formation. The time scale associated withthis process is similar to that of exciton-polaronformation.

• The exciton density of initially prepared quasi-extended states evolves to have peaks correspond-ing to the more localized VRSs, which is consis-tent with the system relaxing onto emissive chro-mophores. Exciton density localization, whose ul-timate cause is disorder, is driven by the externaldissipation through ‘wavefunction collapse’. There-fore, this process has a much longer associated timescale than for exciton-polaron formation and exci-ton decoherence.

We emphasize that one key result of this work isthat exciton decoherence and exciton density localiza-tion, which are terms that are often used interchange-ably in the literature, are shown to be different physicalprocesses that occur on different time scales. While ex-citon decoherence occurs on an intrinsic ultra-fast timescale of ∼10 fs as a result of the vibrational overlaps as-sociated with exciton-polaron formation, exciton densitylocalization (or more precisely, localization of the exciton-polaron itself)36,37 occurs on a much longer extrinsic timescale that is driven by the dissipation to the environmentthrough a ‘wavefunction collapse’ process.

We now attempt to interpret experimental observa-tions in the light of these simulations. As stated in the in-troduction, experimental studies of the exciton relaxationdynamics in PPV observe two sub picosecond time scales(

14

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Appendix A: The Quantum Jump Trajectory Method

The quantum jump trajectory method is a techniquethat allows the numerical solution of a Lindblad mas-ter equation, such as Eq. (6), to be obtained usingtime-dependent Schrödinger equation based methods.More specifically, observables associated with the time-dependent density matrix solution of a Lindblad masterequation are obtained by averaging over many so-calledquantum jump trajectories. The time evolution of a sin-gle quantum jump trajectory is obtained using the fol-lowing procedure:

• Step 1: We first spilt up the total time evolu-tion of a single trajectory into several time stepsof length δt. We next define an effective Hamilto-nian for the system as follows:41

Ĥeff = Ĥ −iγ

2

∑n

b̂†nb̂n (A1)

where Ĥ is the system Hamiltonian, γ is the Lind-

blad master equation dissipation parameter and b̂nis the Lindblad operator for site n. Evolving thissingle quantum trajectory, initially in state |ψ(t)〉,under the effective Hamiltonian gives us a trialstate at time t+ δt:

|ψtrial(t+ δt)〉 = e−iĤeffδt |ψ(t)〉 (A2)

15

• Step 2: As the effective Hamiltonian in Eq. (A1) isnon-Hermitian, the time evolution in Eq. (A2) doesnot conserve the quantum state’s norm:

〈ψtrial(t+ δt)|ψtrial(t+ δt)〉 = 1− δp (A3)

where δp is the amount the norm has decayed overthe time step δt. Using Eq. (A3), the state of thesingle quantum trajectory at time t + δt is deter-mined probabilistically as follows:

a) With probability 1−δp, the state at time t+δtis:

|ψ(t+ δt)〉 = e−iĤeffδt |ψ(t)〉√

1− δp(A4)

b) With probability δp, the state at time t + δtis:

|ψ(t+ δt)〉 = b̂n |ψ(t)〉√〈ψ(t)|b̂†nb̂n|ψ(t)〉

(A5)

where b̂n is the Lindblad operator for site n. Thesite at which the Lindblad operator is appliedin Eq. (A5) is again determined probabilistically,where the probability associated with applying theLindblad operator at site n is given by:

Pn =〈ψ(t)|b̂†nb̂n|ψ(t)〉∑m 〈ψ(t)|b̂

†mb̂m|ψ(t)〉

(A6)

Hence, we now need a technique by which to apply thetime step evolution operator, exp[−iĤeffδt], for the sys-tem of interest. This is achieved for the Frenkel-Holsteinmodel in this work using the TEBD technique, which isintroduced in Appendix B.

Appendix B: Time Evolving Block Decimation

1. Matrix Product States

It has been found that a practical quantum state rep-resentation for one-dimensional systems involves usingmatrix product states (MPS). The MPS representationof a generic quantum state is as follows:22

|ψ〉 =∑

σ1,..,σL

∑α1,..,αL−1

Aσ1α1Aσ2α1,α2 ...A

σLαL−1 |σ1〉⊗ ....⊗|σL〉

(B1)where σn are known as the physical indices, which cor-respond to the quantum numbers of the site basis, |σn〉,and αn are known as the internal indices. For systemsmade up of L sites, with a physical index dimension d, anarbitrary quantum state can be exactly represented by aMPS with an internal index dimension of dL/2. However,most states can be accurately represented with a much

FIG. 14. Tensor network diagram corresponding to a threesite MPS.

smaller internal index dimension, where increased inter-nal index dimensions are required to represent quantumstates with a greater amount of entanglement betweensites.

MPSs can be helpfully represented and manipulatedusing tensor network diagrams.59 Figure 14 shows thetensor network diagram for a three site MPS. In thesediagrams, the circles represent the tensors in the MPS,Aσnαn,αn+1 , while the lines correspond to their indices. Ifa line is ‘closed’, then the index it represents is implicitlysummed over.

2. Trotter Decomposition

In order to obtain the quantum dynamics for a sys-tem, the evolution operator for a time step, exp[−iĤδt],must first be computed. For systems that have a largeHilbert space, this is not possible. Within the TEBDtechnique, this issue is circumvented by using the Trot-ter decomposition to expand the evolution operator intoseveral smaller and more manageable terms.

For Hamiltonians that contain solely on-site and near-est neighbor terms, it can be shown that the full Hamil-tonian can be written as the following sum:

Ĥ =

L−1∑n=1

Ĥn,n+1 (B2)

where Ĥn,n+1 is the Hamiltonian for the “bond” linkingsites n and n + 1 and L is the total number of sites inthe system. Using Eq. (B2), the Trotter decompositionfor the evolution operator is given by:60

e−iĤδt ' e−i2 Ĥ1,2δte−

i2 Ĥ2,3δt...e−

i2 Ĥ2,3δte−

i2 Ĥ1,2δt+O(δt3)

(B3)The Trotter decomposition given in Eq. (B3) is not ex-

act, as the Hamiltonian terms Ĥn,n+1 in general do notcommute with each other. However, the error associatedwith this Trotter decomposition is of O(δt3), which be-comes negligible if the time steps are chosen to be small.

3. The Procedure

The procedure for applying a single Trotter decom-position evolution operator onto a MPS is illustrated in

16

Step 1

Step 2

Step 3

FIG. 15. Diagram outlining the TEBD procedure.

Fig. 15, with the first tensor network diagram in this fig-ure corresponding to the object exp[−iĤn,n+1δt/2] |ψ〉.This figure illustrates the following steps:

• Step 1: The closed indices αn, σn and σn+1 aresummed over to form the new tensor Θ

σ̃n,σ̃n+1αn−1,αn+1 .

From Fig. 15, we see that this new tensor is givenby:23

Θσ̃n,σ̃n+1αn−1,αn+1 =∑

σn,σn+1,αn

U σ̃n,σ̃n+1σn,σn+1Aσnαn−1,αnA

σn+1αn,αn+1

(B4)where

U σ̃n,σ̃n+1σn,σn+1 = 〈σ̃n, σ̃n+1| e− i2 Ĥn,n+1δt |σn, σn+1〉 (B5)

• Step 2: A singular value decomposition is appliedon the tensor Θ

σ̃n,σ̃n+1αn−1,αn+1 to obtain three new ten-

sors, as well as a new internal index α̃n:23

Θσ̃n,σ̃n+1αn−1,αn+1 = Bσ̃nαn−1,α̃n

Sα̃n,α̃nVσ̃n+1α̃n,αn+1

(B6)

If the size of the original internal index αn was χ,then the new internal index α̃n has grown to sizeχd on completion of the procedure outlined above,where d is the size of the site basis. To keep the in-ternal indices constant in size, we truncate the α̃nindex by discarding the α̃n ‘states’ associated withthe smallest values of Sα̃n,α̃n .

23 The error associ-ated with this truncation procedure can be quanti-fied by computing the sum of the discarded singularvalues, Sα̃n,α̃n , which gives a bound on the error as-sociated with the time evolved quantum state. Itis this truncation procedure that keeps the totalHilbert space of the system finite and manageablethroughout the dynamics.

• Step 3: The resulting tensor network diagram istransformed back into MPS form by creating the

new tensor Bσ̃n+1α̃n,αn+1

:

Bσ̃n+1α̃n,αn+1

= Sα̃n,α̃nVσ̃n+1α̃n,αn+1

(B7)

Hence to complete a dynamics time step within theTEBD algorithm, the same three step procedure outlinedabove is performed for every evolution operator in theTrotter decomposition, given in Eq. (B3).

Outlined above is the necessary framework for numer-ically obtaining the dynamics associated with the Lind-blad master equation given in Eq. (6). For our simula-tions of the exciton relaxation dynamics for PPV polymerchains containing 99 moieties/sites, the parameters usedin the TEBD algorithm are given in Table III. These wereobtained as follows. The maximum number of phononsper site, nphon = 2, was chosen as it gives physically sensi-ble results for the dynamics, while keeping the size of thesite Hilbert space manageable. The simulations were alsoperformed for nphon > 2, where we found that the resultsdid not deviate significantly from the nphon = 2 calcula-tions. Using dimensionless time, t̃ = ωt, the value ofthe time step, δt̃, was obtained by comparing the TEBDresults for a four site Frenkel-Holstein model with thenumerically exact dynamics obtained from exact diago-nalization. As for the case of determining nphon, we choseto use χ = 150 as a balance between obtaining accurateresults for the quantum dynamics, as well as making thecalculation not too computationally intensive. The errorassociated with this Hilbert space truncation (quantifiedfrom the sum of the discarded singular values, Sα̃n,α̃n ,obtained during the singular value decompositions withinthe TEBD algorithm) was found to be small for each timestep during our dynamics simulations, suggesting that wecan be confident that the Hilbert space truncation doesnot lead to a large error in our results for these parametervalues.

Parameter Value Parameter Value

δt̃ 5× 10−4 χ 150nphon 2

TABLE III: Parameters used in the time evolving blockdecimation technique (TEBD).

Ultra-Fast Relaxation, Decoherence and Localization of Photoexcited States in -Conjugated Polymers: A TEBD StudyAbstractI IntroductionII Polymer Conformations and Coarse-GrainingIII Models and Numerical TechniquesA The Frenkel ModelB The Frenkel-Holstein ModelC The Lindblad Master EquationD Optical IntensityE Numerical Techniques

IV ResultsA Exciton-Polaron FormationB Exciton DecoherenceC Exciton Density LocalizationD LEGSs DynamicsE Time Resolved Fluorescence Anisotropy

V Discussion and ConclusionsVI AcknowledgementsA The Quantum Jump Trajectory MethodB Time Evolving Block Decimation1 Matrix Product States2 Trotter Decomposition3 The Procedure