## Abstract

Symmetric couplings among aggregates of *n* chromophores increase the transfer rate of excitons by a factor *n*^{2}, a quantum-mechanical phenomenon called ‘supertransfer’. In this work, we demonstrate how supertransfer effects induced by geometrical symmetries can enhance the exciton diffusion length by a factor *n* along cylindrically symmetric structures, consisting of arrays of rings of chromophores, and along spiral arrays. We analyse both closed-system dynamics and open quantum dynamics, modelled by combining a random bosonic bath with static disorder. In the closed-system case, we use the symmetries of the system within a short-time approximation to obtain a closed analytical expression for the diffusion length that explicitly reveals the supertransfer contribution. When subject to disorder, we show that supertransfer can enhance excitonic diffusion lengths for small disorders and characterize the crossover from coherent to incoherent motion. Owing to the quasi-one-dimensional nature of the model, disorder ultimately localizes the excitons, diminishing but not destroying the effects of supertransfer. When dephasing effects are included, we study the scaling of diffusion with both time and number of chromophores and observe that the transition from a coherent, ballistic regime to an incoherent, random-walk regime occurs at the same point as the change from supertransfer to classical scaling.

## 1. Introduction

In 1954, Dicke introduced the phenomenon of *superradiance*, a quantum interference effect induced by symmetries of spin–boson interactions, in which many interacting atoms could collectively conspire to yield an enhanced relaxation rate [1]. Owing to this fundamental non-classical cooperation, the probability of a single photon emission from *n* identical atoms collectively interacting with vacuum fluctuations becomes *n* times larger than incoherent individual spontaneous emission probabilities [2]. Superradiant radiative relaxation can also occur in molecular aggregates owing to the inherent coherent feature of Frenkel exciton dynamics. This phenomenon can be observed when a closely packed group of molecules interacting under certain symmetry can collectively donate an excitation with a rate that is much faster than each individual molecule. The adversarial effects of inhomogeneous broadening and exciton–phonon interactions on such cooperative relaxation in molecular systems have been studied in detail [3–6].

The same symmetry principles that underlie superradiance can give rise to an analogous phenomenon known as cooperative energy transfer or *supertransfer* [7–9]. Generally, the exciton transfer rate can be calculated from the transition probability of an excitation jumping from one molecule to another using fluorescence resonance energy transfer (FRET), based on a perturbation treatment of the dipole–dipole interaction between individual molecules. However, under symmetrized interactions of a group of *n* molecules, the excitation becomes highly delocalized, leading to a large (effective) dipole moment associated with the entire group. The resulting enhanced oscillator strength can lead to supertransfer when similar molecular assemblies, with comparable effective dipole moments, exist that can play the role of acceptors. Under such conditions, the rate of exciton dynamics should be calculated from these effective large dipole–dipole interactions to describe the coherent donation and acceptance among such molecular aggregates, with up to *n*^{2} enhancement over the single molecule to single molecule transfer rate, even in the far field [9]. A primary goal of this paper is to study the behaviour of superradiance, supertransfer and other non-classical collective phenomena in the presence of disorder and environmental fluctuations similar to those natural conditions of photosynthetic light-harvesting complexes [3–5,8,10]. Recently, the existence and role of quantum coherence in the dynamics of excitation energy transfer in biological systems have been of significant interest both experimentally [11–17] and theoretically [18–33]. Such studies can provide novel concepts and techniques for a deeper understanding of natural/engineering excitonic systems, potentially leading to efficient and robust artificial light-harvesting [31,32].

A major problem in the design and fabrication of novel excitonic devices is the limited exciton diffusion length, which could be about 10 nm in disordered materials. This issue has led to low efficiency and complicated device structures in organic photovoltaic cells [34,35], and it is a bottleneck in the performance of excitonic transistors [36] and the triplet–triplet energy transfer in organic light-emitting diodes [37]. A key open question is whether one can use quantum-mechanical supertransfer effects to enhance exciton diffusion length in such disordered systems. Recent experimental investigations of nano-engineered biological systems suggest that under laboratory conditions certain aspects of photosynthetic complexes can be emulated that could be potentially exploited for efficient energy transport [38,39].

Self-assembled ring structures containing fluorescent chromophores attached to tobacco mosaic virus coat monomers exhibit efficient exciton transport together with a broad-spectrum light collection with over 90 per cent efficiency [39]. It is of considerable interest to explore whether the energy transport mechanism in such systems was facilitated in part due to symmetries in the arrangement of chromophores, which could give rise to the phenomenon of supertransfer [7–9].

In this paper, we investigate excitonic transport in systems consisting of rings of chromophores stacked in cylindrical arrays, as a function of the number of chromophores per ring, the spacing between rings and the strength of decoherence and disorder. We also investigate excitonic transport in dipole-coupled spiral structures. Such geometries are relevant not only to the experimentally investigated systems mentioned above, but also to naturally occurring cylindrical arrays of chromophores such as the green sulfur bacterium chlorosome. We use the symmetries of the system to derive analytical solutions for the behaviour of the closed system in the absence of environmental interactions and perform simulations to capture the dynamics of excitonic diffusion in the presence of environmentally induced noise and disorder. Our results provide clear evidence for the presence of supertransfer in the appropriate regimes and for the destruction of supertransfer in other regimes.

Specifically, we study the effect of supertransfer on the diffusion length *σ* of an initially delocalized exciton along a linear chain of chromophoric rings and helical rods, mediated by dipolar couplings between chromophores. Supertransfer-induced enhancements in the hopping rate translate into a commensurate increase in the distance travelled by the excitons. The strength of the supertransfer effect depends on the number of chromophores per ring *n* (higher *n* yields higher supertransfer rates), and on the distances between rings (smaller inter-ring distances yield larger asymmetries and diminishes supertransfer). By changing the strength of the interaction with the environment, we investigate excitonic transport in both coherent/ballistic and incoherent/diffusive regimes. By changing the degree of disorder, we investigate how the degree of localization depends on *n*. The analysis is carried out analytically in the closed-system case, and numerically in the presence of disorder and interactions with a bosonic bath. Our analytical and numerical solutions allow us to investigate scenarios where rings are packed closely in a cylinder, and where chromophores are arranged in spirals.

## 2. Theoretical model

### (a) A simple example

The concept of supertransfer can be easily reviewed in the case of two-spin systems *A* and *B*, containing *n*_{A} and *n*_{B} sites, respectively, invariant under permutation symmetry [9]. In the presence of such symmetric couplings, the hopping rate from the symmetrized single-excitation state of *A* to the symmetrized single-excitation state of *B* is *n*_{A}*n*_{B} times the hopping rate of a localized excitation from one of the *A* to one of the *B* sites. (The effect can be even larger—up to —for multiple excitation states [9]. For simplicity, this paper will focus on single-exciton states). Consider the following example, where supertransfer is captured in a simple way by the following symmetric hopping Hamiltonian:
2.1
Here, *i* labels the *n*_{A} sites, and *j* the *n*_{B} sites of *A* and *B*, respectively. Note that all sites interact via a unique coupling *γ*, making the dynamics of the excitons highly symmetrical and supertransfer possible.

Introducing the angular momentum operators
2.2
the previous Hamiltonian can be expressed as
2.3
From the above expressions, the probability amplitude for a single symmetrized excitation state |*ϕ*_{A}〉 over the system *A*, , to hop to a corresponding symmetrized state over system *B* is given by , in first-order perturbation theory. The corresponding transition probability *γ*^{2}*n*_{A}*n*_{B} is therefore *n*_{A}*n*_{B} times the probability *γ*^{2} for a localized exciton state |1〉 to hop from one of the *A* sites to any one of the *B* sites. This enhancement effect in the hopping rate is called supertransfer [7–9]. The key feature of supertransfer is coherence within the individual systems *A* and *B*, respectively. Supertransfer between *A* and *B* can be either coherent or incoherent, depending on the spatial separation of the systems and the strength of the interaction with the environment.

### (b) Exciton supertransfer in cylindrical geometries

Here, we study supertransfer effects in cylindrical aggregates of sites interacting via a 1/*r*^{3} potential. We apply our analysis to the transition dipoles of an array of chromophores to look at excitonic hopping through the array. We consider diffusion in the context of two types of geometrical set-ups. In the first configuration, the sites are arranged in rings, which are stacked in a cylinder. Each ring has a radius *R*, and consists of *n* dipoles each. There are a total of *N* rings, arranged co-axially in a cylinder with a separation *D* between adjacent rings. The position of the *i*th site inside the *j*th ring is given by , for *i*=1,2,…,*n* and *j*=1,2,…,*N*. In the second geometry, chromophores form helical rods, with their positions given by , where *d* is the pitch of the helix, *R* its radius, *n* the number of chromophores per turn, with *N* turns in total, and *i*=1,2,…,*nN*. Figure 1 shows the different geometries and labels employed.

The single-exciton manifold approximation will be employed, so that the system Hilbert space is , spanned by a basis , where |*m*〉 denotes the state in which the *m*th molecule is excited.

The interaction among all chromophores in the system will be modelled by the following Hamiltonian:
2.4
with all the interactions included. For simplicity, we initially consider an interaction of the form , with *r*_{mn}=|*r*_{m}−*r*_{n}|. In what follows, the normalization *J*=1 will be adopted, with time measured in units of 1/*J*. This interaction is a simplified version of the dipolar interaction that has the same distance dependence as the conventional dipolar interaction, but does not take into account the orientations of the dipoles. This simplified assumption will be relaxed below.

The influence of the environment over the system will be considered by including both on-site energy disorder and on-site dephasing (Haken-Strobl model), which has recently been widely used for studying environmental effects on energy transport in light-harvesting systems [20,23,25]. In (2.4), 〈*E*〉 represents the average molecular excitation energy, which can be dropped by shifting all the energies by 〈*E*〉. The static inhomogeneous offset *η*_{r} in the energy of the *r*th site reflects the disorder caused by the surroundings. We take *η*_{r} to be a Gaussian random variable with standard deviation *Σ* and probability distribution given by
2.5
where we ignore correlations between the offsets of each molecule. The strength of the disorder *Σ* is measured in units of *J*.

The effects of the bosonic bath surrounding the system are modelled by an on-site dephasing model or Haken–Strobl model, given by a Lindblad superoperator of the form
2.6
with the sum running over all the sites, *S*_{n}=|*n*〉〈*n*| and *γ* the dephasing rate. Thus, the total dynamics is given by
2.7
where the effects of exciton recombination are captured by the final term. Here, , with 1/*κ* the lifetime of the exciton.

### (c) Choice of initial state and diffusion length

By its definition, supertransfer relies fundamentally on two different symmetries: the permutation symmetry present on the Hamiltonian, as well as the permutation symmetry and coherence on the initial quantum states. Both the ring chains and helical rods set-ups resemble the structures of the fluorescent chromophores attached to tobacco mosaic virus monomers, which can self-assemble into stacks of rings or helices [39]. Note that the invariance under permutations of the *n* dipoles inside a ring resembles the symmetric interaction in (2.1) for *D*>*R*, where strong supertransfer effects should take place. When the rings become closer, *D*<*R*, then the interaction between rings becomes less symmetric. The strength of supertransfer is diminished, but the effect persists, as we will show analytically below.

As the initial quantum state in the equations of motion, we choose a delocalized quantum state over the ring of *n* sites, of the form *ρ*_{deloc}=|*ϕ*〉〈*ϕ*|, with , supported on the middle (*N*/2) ring of the *N*-ring chain. We consider an odd number *N* of rings, each with an odd number of sites *n*, with *N*=2*T*+1, *n*=2*t*+1. The rings are labelled by *r*=−*T*,…,−1,0,1,…,*T*, with 0 denoting the middle ring. (The use of odd numbers of rings and sites is simply for the convenience of labelling.) To quantify how much an initial excitation located at the middle ring diffuses towards the boundaries, we solve for *ρ*(*t*) in the master equation (2.7) and calculate the second moment *σ*(*t*) of the probability distribution *p*_{i}(*t*) for an exciton to be at the *i*th ring at time *t*:
2.8
with the probability for the exciton to be present in the *r*th ring at time *t*, the exciton population of site *j* on ring *r* and *D* the distance between two adjacent rings. The diffusion length *σ*(*t*) measures how far the exciton diffuses along the system of rings. We begin by presenting our analytical model for the closed quantum system to show that supertransfer enhances the diffusion of a delocalized initial state by a factor *n*. This enhancement for the closed system (coherent evolution and ballistic transport) will be found to persist in the open system, at least for short times.

## 3. Closed-system dynamics: analytical calculation of diffusion

We will proceed to calculate *σ*(*t*) for a closed system using a short-time approximation, that is *t*∥*H*∥≪1, with ∥*H*∥ the operator norm of the Hamiltonian *H* (maximum eigenvalue of *H*). We assume a long lifetime 1/*κ*≫*t*, such that recombination effects can be ignored. An important observation is that the Hamiltonian of the linear chain of rings is very close to a block circulant structure. A block circulant matrix is of the form
3.1
where the *n*×*n* matrices **b**_{i} are themselves circulant, that is *b*_{i}=circ(*b*_{i,0} *b*_{i,1} … *b*_{i,n−1}) [40]. In this context, **b**_{0} represents the Hamiltonian for a single ring, while the matrices **b**_{i}, *i*≠0, represent the *n*×*n* set of interactions between two rings. The circularity of **b**_{0} is ensured by the symmetry of the ring structure. A symmetric, block circulant matrix *B* has the property that **b**_{i}=**b**_{N−i}, *i*=0,1,…,*N*−1. This property would hold exactly if we were to impose periodic boundary conditions along the chain, so that the system is topologically a ‘ring of rings’ or torus. The actual chain Hamiltonian *H* does not possess periodic boundary conditions. Nevertheless, for times short enough so that boundary effects can be neglected, we will see that the analytical expression for diffusion *σ* obtained assuming block circularity matches the numerical results quite well.

A matrix if and only if it commutes with the unitary matrix , where *π*_{N} is the fundamental *N*×*N* permutation matrix *π*_{N}=circ(0 1 … 0), and is the *n*×*n* identity matrix.

Let us then consider the matrix as being block circulant, with circulant blocks. Such a Hamiltonian will have the form *H*=circ(**H**_{0} **H**_{1} …**H**_{T} **H**_{T} … **H**_{1}), where . The spectrum of such matrices is well known [41]. Let and . Define the matrix *F*_{N}(*ω*) given by
3.2
Then, the circulant matrix *H* can be diagonalized by *F*_{N}(*ω*)⊗*F*_{n}(*ρ*), with eigenvalues *e*(*p*,*q*) given by
3.3
3.4
that is, it is the double discrete Fourier transform of the coefficients *h*_{j,k}, where the index *j* labels a block or a ring, and *k* the element inside the *j*th block. For *H* real and symmetric, its real eigenvalues *e*(*p*,*q*) become *T* doubly degenerate for *p*=1,2,…,*T*, and *t* doubly degenerate for *q*=1,2,…,*t*, and are of the form . The corresponding complex conjugated eigenvectors can be combined into two real eigenvectors. With this representation of the eigenvectors, the expression for the diffusion *σ* for a finite chain of rings, for an initial delocalized state, is given by
3.5
where *D* is the spacing between the rings. For a localized state, the corresponding expression for the diffusion is
3.6
These expressions are valid for small times or interaction strengths, that is, *t*∥*H*∥≪1. The proof is given in the appendix, for any initial condition. A couple of simple cases will help us understand these expressions.

In the ‘far-field’ regime, with *D*>*R*, the couplings among the rings will be highly symmetric, similar to (2.1). For this case, **b**_{1}=**b**_{N−1}=*V* *J*_{n}, with the rest of the block matrices **b**_{i}=0, where *J*_{n}=circ(1 1 … 1) is an *n*×*n* matrix with ones as entries and *V* =*J*(1/*D*_{nm})^{3} is the strength of the symmetric couplings between two nearest-neighbouring rings, separated by a distance *D*. The expressions above for the diffusion will read and , so that the symmetric couplings between the rings produce a linear scaling of the diffusion as a function of the number of sites *n* and for delocalized and localized initial states, respectively. Thus, we have proved that a linear dependence with the number of particles *n* occurs under symmetric couplings, and will be used in the rest of this work as a proof of supertransfer. We also obtain that for a 1/*r*^{3} dipole–dipole decaying interaction, the diffusion decreases as 1/*D*^{2}, with *D* the distance between rings, and is directly proportional to the typical interaction strength *J*. If all the interactions among the rings were included, the above expressions change to and .

In the ‘near-field’ regime, with *D*≤*R*, the scaling with *n* can increase beyond *n* or for *σ*_{deloc} and *σ*_{loc}, respectively. The reason comes from how the interaction terms and scale up with *n*. Without loss of generality, let us consider two nearest-neighbouring rings. The interaction term for the delocalized case is of the form , which exhibits a scaling with *n* faster than linear from *D*≤*R*, for fixed *j*. This is a small size effect, and the scaling is linear for *n* large enough (as can be readily checked by approximating the above sum by an integral for *n*≫1).

The expression for the diffusion of a delocalized state (3.5) shows also some interesting ‘interference effects’, in the sense that partial cancellations can occur when some of the coefficients *h*_{j,k} are negative. In particular, when the condition is met, the delocalized state does not propagate, while the localized state does. Therefore, even in the closed system, an initial delocalized state does not necessarily diffuse longer than a localized state, during times *t*∥*H*∥≪1.

Figure 2 makes a comparison between the simulated value of *σ*(*t*)/*Dt*, for delocalized and localized initial states, and for various values of the distance *D* between the rings.

In the far-field regime *D*>*R*, we clearly see a linear scaling of diffusion with *n* for a delocalized initial state, and with for a localized state. For the near-field regime *D*<*R*, there is a superlinear scaling that arises from the behaviour of the interaction terms in the Hamiltonian, as mentioned earlier. In all cases, there is a good agreement between the theoretical predictions and the numerical simulations. For times much longer than 1/∥*H*∥, the excitons travel long enough so that the effects of the boundaries ‘kick in’, and the block circularity assumption worsens.

According to equations (3.5) and (3.6), the diffusion does not depend on the set of inner ring interactions, since their label *j*=0 does not contribute to the diffusion. Therefore, provided *t*∥*H*∥≪1 and the block circularity approximation holds, the inner ring couplings are irrelevant for diffusion, for both the localized and delocalized cases. We have confirmed this prediction by numerical simulations, where the coupling strength of the inner rings were either increased or decreased by the same amount, keeping the inter-ring couplings *J*=1 unchanged. Equations (3.5) and (3.6) correctly predict the observed diffusion, under the above approximations.

The above analysis that was developed for a stacked ring geometry can be used to approximate the diffusion in the helical rod setting, if the helices in the rod are approximated by stacked rings of the same radius *R*, with a distance between them equal to the pitch of the helix *d*=*D* and with number of chromophores per ring *n* equal to that in the rods per turn. Figure 3 exhibits this approximation. The blue circles correspond to numerical simulations for the spiral geometry with given *R* and pitch *d*, while the red stars depict the formulae (3.5) and (3.6) obtained from a set of facing rings with radius *R* and separation *D*=*d*, indicating that the approximation given by the ring structure is quite good. In principle, if the number of chromophores per turn is not an integer, the angular position of the rings should be shifted by a given amount to account for this. However, owing to the parametrization of the spiral that is being used, the nodes are perfectly aligned along the axis of the spiral, so no ‘shift’ of the nodes positions inside the rings is necessary.

## 4. Open-system dynamics: numerical simulations

Having elucidated the behaviour of diffusion length in a closed-system dynamics, in this section, we analyse the behaviour of supertransfer subjected to an open-quantum-system dynamics, via two effects: on-site energy disorder and on-site dephasing.

### (a) Effects of on-site disorder

For small values of static energy disorder, the motion of the exciton should be hardly affected, propagating coherently. For times *t*≫*Σ*^{−1}, diffusive behaviour sets in and the mismatch among the on-site energies causes localization [42]. This is depicted in figure 4. For disorder *Σ*≪1/*t*, the motion is coherent, and we obtain a supertransfer regime *σ*∼*n*, while the exciton gets localized for *tΣ*≫1. A similar conclusion can be drawn from figure 5, where the exponent *λ* in *σ*(*t*)∼*t*^{λ} shows a coherent (*λ*=1) evolution for small values of disorder, and a crossover to classical diffusion takes place when *tΣ*∼1, independently of *n*. In the classical regime, the diffusion fluctuates around *λ*=0.5.

Figure 6 represents a level curve for diffusion as a function of *n* and disorder *Σ*, obtained from figure 4. For disorder *Σ*<10^{−2}, the system is resilient to changes of disorder, and we see a linear enhancement of diffusion with *n*, while for *Σ*>10^{−2}, diffusion has a behaviour. The areas between the different values of disorder provide an indication of how robust diffusion is under changes of disorder. From this observation, one derives that diffusion seems more robust under changes of disorder in the quantum or ballistic regime (for *Σ*<10^{−2}) than in the diffusive regime.

### (b) Effects of on-site dephasing

On-site dephasing diminishes the coherence among the sites of the *i* ring, which is fundamental for supertransfer effects to take place. For a tight-binding Hamiltonian on an infinite chain with periodic boundary conditions, nearest-neighbour interaction *J* and dephasing strength *γ*, see [43], the diffusion length is given by
4.1
For small times, *γt* ≪ 1, the transport is initially ballistic, *σ*(*t*) ∼ *t*, while for *γt* ≫ 1, the transport is diffusive, , with a crossover time of the order of 1/*γ*.

A similar behaviour should be expected for the diffusion for the ring structure. For small times compared with this dephasing scale, *t*<1/*γ*, supertransfer effects will enhance the diffusion length by a factor of *n*. This is shown in figure 7, for *γ*=1, where at small times, the motion is still coherent and *σ*(*t*,*n*)∼*n*, while for *t*>1/*γ*, the motion is classical and .

Using the block-circulant approximation employed in the closed-system dynamics from the previous section, a similar calculation for diffusion can be carried out in the presence of dephasing as well. It is possible to verify that dephasing does not contribute to the diffusion, up to second order in the expansion of , with , for *ρ*(0) being either a localized or delocalized state. That is, up to second order in time, the expressions (3.5) and (3.6) remain unmodified under dephasing. Figure 8 shows the diffusion as a function of the number of nodes *n*, for an initial delocalized state (upper panels) and localized state (lower panels), in the near-field *D*/*R*=0.1, *D*/*R*=1 and far-field *D*/*R*=10 regimes. Most importantly, the theoretical predictions (3.5) and (3.6) (blue circles) are contrasted with numerical simulations of diffusion including many values of dephasing: *γ*=0.1 (red triangles), *γ*=1 (green diamonds) and *γ*=10 (brown squares). In all cases, for small values of dephasing *γ*=0.1, the agreement between the numerical and analytical results is quite good, and the relative error between them is at most 3 per cent for all cases.

Figure 9 shows the best fit for the exponent *λ* in *σ*(*t*)∼*t*^{λ}, for a fixed value of dephasing and different number of sites *n*. The exciton starts initially with a ballistic spreading *λ*=1, decaying to a diffusive classical walk regime *λ*=0.5. As the number of sites increases, we see a sharper drop of the exponent towards 1/2. The dependence with *n* comes from the fact that the decoherence rate of a symmetric state coupled symmetrically with a common on-site dephasing bath goes as *n* times the single-exciton rate *γ*. The dip in the scaling exponent *λ* below the classical value 0.5 around *t*=1 may be due to suppression of symmetry due to dephasing. Indeed, dephasing destroys the coherence of the density matrix, and, therefore, the information about the circular symmetry of the ring system. This can lead to some destructive interference and partial localization, suppressing the overall transfer rate.

A similar analysis for the scaling *λ* was performed in Hoyer *et al*. [26] for the exciton diffusion in the Fenna–Matthews–Olson (FMO) protein complex. They observe a ‘subdiffusive’ regime, i.e. *λ*<0.5, that they attribute to the interplay between disorder and decoherence. However, their analysis does not apply to our case, as the results in figure 9 are obtained under the effects of decoherence alone, with no disorder. We believe that this effect is inherently a multi-chromophore effect within the rings, as it arises only for *n*>2.

The scaling *α* in *σ*(*t*)∼*n*^{α} as a function of time, for different values of dephasing, is shown in figure 10. The scaling goes from the supertransfer *α*=1 to the classical walk regime *α*=1/2 around the same time that exciton diffusion starts to deviate from the ballistic spreading *λ*=1.

## 5. Conclusions

We have analysed both analytically and numerically how symmetry-enhanced supertransfer, for circular and helical geometry of chromophores, enhances excitonic diffusion lengths. For a closed-system dynamics, we derived explicit expressions for diffusion for any initial state, demonstrating a factor *n* enhancement in diffusion when the inter-ring couplings are similar to each other. These formulae explicitly capture quantum interference effects for initial delocalized states, independently from the details of the rings where the excitons are initially created, and approximate the diffusion along helical rods quite well.

Moreover, we have studied the effects of environmental interactions on supertransfer, by including both energy disorder and interactions with a bosonic bath in a Haken–Strobl pure dephasing model. The *n* enhancement in diffusion prevails for times smaller than the disorder strength, and the system becomes more resilient against random perturbations in the on-site energies for increasing values of the number of nodes *n* per ring or turn of the spiral. Owing to the quasi-one-dimensional nature of the aggregates considered here, supertransfer effects are fragile to disorder on the site basis energies. We have shown analytically that the dephasing does not affect the form of our closed expressions for diffusion, up to second order in time, in good agreement with our simulations. A numerical analysis enabled us to obtain the scaling exponents of *σ* with number of nodes *n* and time *t*, revealing a crossover from the ballistic supertransfer regime, to a diffusive normal regime, in both cases around the same characteristic time 1/*γ*. Our studies on exciton supertransfer dynamics presented here can be generalized for other complex quantum systems interacting with non-Markovian and non-perturbative environments using the new techniques that have recently been developed in Shabani *et al*. [31] and Mohseni *et al*. [32].

## Acknowledgements

The numerical simulations described in this paper were supported by the University of Southern California Center for High Performance Computing and Communications. We acknowledge financial support from DARPA under the QuBE program and ENI (M.M., S.L.). S.L. acknowledges support of NEC, Intel, NSF, ONR, ISI, and the Santa Fe Institute. P.Z. acknowledges support from NSF grants PHY-803304, DMR-0804914 and PHY-0969969.

## Appendix A. Derivation of the closed expressions for the exciton diffusion length

As shown in (3.3), the eigenvalues of a block circulant matrix are given by
Assume the initial state is given by
where, for simplicity, we assume *α*_{r,s} to be real. The second moment or diffusion *σ*^{2}(*t*) is given by
where
is the probability that the exciton is in the ring *R*. Using the spectral decomposition of the evolution operator , we obtain
where we have used that the eigenvectors have the tensor product structure |*e*(*j*,*k*)〉=|*e*(*j*)〉|*e*(*k*)〉. For *t*∥*H*∥≪1, we employ the approximation . Putting all the terms together and expanding up to second order in time, we get
where we have used the convention that *α*_{−R,S}=*α*_{R,−S}=*α*_{R,S}.

For an initial delocalized state on the 0 ring, and
Therefore, the diffusion is given by
For a localized state on the ring 0 and site 0, we have *α*_{R,S}=*δ*_{R,0}*δ*_{S,0} and
and the diffusion is given by

## Footnotes

One contribution of 14 to a Theo Murphy Meeting Issue ‘Quantum-coherent energy transfer: implications for biology and new energy technologies’.

- This journal is © 2012 The Royal Society