Opening-Assisted Coherent Transport in the Deep Classical Regime
OOpening-Assisted Coherent Transport in the Deep Classical Regime
Yang Zhang, G. Luca Celardo,
2, 3, 4
Fausto Borgonovi,
2, 3 and Lev Kaplan Department of Physics and Engineering Physics,Tulane University, New Orleans, Louisiana 70118, USA Dipartimento di Matematica e Fisica and Interdisciplinary Laboratories for Advanced Materials Physics,Universit`a Cattolica del Sacro Cuore, via Musei 41, I-25121 Brescia, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico (Dated: September 20, 2018)We study quantum enhancement of transport in open systems in the presence of disorder and de-phasing. Quantum coherence effects may significantly enhance transport in open systems even in thedeep classical regime (where the decoherence rate is greater than the inter-site hopping amplitude),as long as the disorder is sufficiently strong. When the strengths of disorder and dephasing are fixed,there is an optimal opening strength at which the coherent transport enhancement is optimized.Analytic results are obtained in two simple paradigmatic tight-binding models of large systems: thelinear chain and the fully connected network. The physical behavior is also reflected in the FMOphotosynthetic complex, which may be viewed as intermediate between these paradigmatic models.
PACS numbers: 71.35.-y, 72.15.Rn, 05.60.Gg
I. INTRODUCTION
Since the discovery that quantum coherence may have a functional role in biological systems even at room tem-perature [1–5], there has been great interest in understanding how coherence can be maintained and used under theinfluence of different environments with competing effects. In particular, much recent research has focused on quan-tum networks, due to their relevance to molecular aggregates, such as the J-aggregates [6], natural photosyntheticsystems [7], bio-engineered devices for photon sensing [8], and light-harvesting systems [9].Many photosynthetic organisms contain networks of chlorophyll molecular aggregates in their light-harvesting com-plexes, e.g.
LHI and LHII [10]. These complexes absorb light and then transfer the excitations to other structures orto a central core absorber, the reaction center, where charge separation, necessary in the next steps of photosynthe-sis, occurs. Exciton transport in biological systems can be interpreted as an energy transfer between chromophoresdescribed as two-level systems. When chromophores are very close, which for chlorophylls is often less than 10 ˚A,the interaction between them is manifested in a manner known as exciton coupling. Under low light intensity, inmany natural photosynthetic systems or in ultra-precise photon sensors, the single-excitation approximation is usu-ally valid. In this case the system is equivalent to a tight binding model where one excitation can hop from site tosite [7–9, 11–13].Light-harvesting complexes are subject to the effects of different environments: i ) dissipative, where the excitationcan be lost; and ii ) proteic, which induce static or dynamical disorder. The efficiency of excitation transfer can bedetermined only through a comprehensive analysis of the effects due to the interplay of all those environments.Here we consider systems subject to the influence of a single decay channel, in the presence of both static anddynamical disorder. The decay channel represents coupling to a central core absorber (loss of excitation by trapping).For many molecular aggregates, the single-channel approximation is appropriate to describe this coupling, modeledfor instance by a semi-infinite one-dimensional lead [14–16]. The disorder is due to a protein scaffold, in whichphotosynthetic complexes are embedded, that induces fluctuations in the site energies. Fluctuations that are slow orfast on the time scale of the dynamics are described as static or dynamic disorder, respectively.Several works in the literature aim to understand the parameter regime in which transport efficiency is maximized.Some general principles that might be used as a guide to understand how optimal transport can be achieved have beenproposed: Enhanced noise assisted transport [17, 18], the Goldilocks principle [19], and superradiance in transport [20].It is well known that when a quantum system is strongly coupled to a decay channel, superradiant behavior mayoccur [21]. Superradiance implies the existence of some states with a cooperatively enhanced decay rate, and is alwaysaccompanied by subradiance, the existence of states with a cooperatively suppressed decay rate. Though originallydiscovered in the context of atomic clouds interacting with an electromagnetic field [22], and in the presence of manyexcitations, superradiance was soon recognized to be a general phenomenon in open quantum systems [21] under theconditions of coherent coupling with a common decay channel. Crucially, it can occur in the presence of a singleexcitation (the “super” of superradiance [23]), entailing a purely quantum effect. Most importantly for the presentwork, superradiance can have profound effects on transport efficiency in open systems: for example, in a linear chain,the integrated transmission from one end to the other is peaked precisely at the superradiance transition [14]. a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug ω N ω ω ω N - ω Ω Ω Ω Sink Γ trap Sink Γ trap Ω Ω Ω Ω Ω Ω ω ω ω ω FIG. 1: (Color online) Upper panel: The linear chain. One excitation can hop between N sites, with on-site energies ω i andwith nearest neighbors connected by tunneling transition amplitude Ω. Site N is connected to a decay channel, where theexcitation can escape, with coupling strength Γ trap . Lower panel: The analogous fully connected model with N = 4 sites. The functional role that superradiance might have in natural photosynthetic systems has been discussed in manypublications [5, 8, 24, 25], and experimentally observed in molecular aggregates [6, 26]. Superradiance (or supertrans-fer) is also thought to play an important role in the transfer of excitation to the central core absorber [5], and its effectson the efficiency of energy transport in photosynthetic molecular aggregates have recently been analyzed [20, 27].While superradiance may enhance transport, static disorder is often expected to hinder it, since it induces localiza-tion [28]. The relation between superradiance and localization has been already analyzed in the literature in differentcontexts [13, 29–31]. Additionally, dynamical disorder (or dephasing) will generally destroy cooperativity [32], andhence counteract quantum coherence effects, including superradiance. On the other hand, dynamical disorder mayalso enhance efficiency, through the so-called noise assisted transport [17, 18].In the deep classical regime where dephasing is stronger than the coupling between the chromophores, transport inquantum networks can be described by incoherent master equations with an appropriate choice of transition rates.However, the presence of an opening (trapping) introduces a new time scale to the system. When the opening strengthis large, coherent effects may be revived even in the deep classical regime. Here we want to address the followingquestions: i ) For which values of the opening strength are coherent effects relevant? ii ) Can we enhance transport byincreasing the opening, which induces coherent effects not present in the incoherent model? iii ) Under what genericconditions can coherent effects enhance transport in open quantum systems?The remainder of the paper is organized as follows. In Sec. II, we present the basic mathematical formalism foranalyzing the dynamics of open quantum networks in the presence of both static disorder and dephasing, and definethe average transfer time, which measures the transport efficiency in these systems. Then in Sec. III, we first focus onthe two-site model, where all results may be obtained analytically, and determine the regime of dephasing, detuning,and opening strength in which quantum coherent effects enhance quantum transport. Specifically, we show that ifthe strengths of static and dynamical disorder (detuning and dephasing, respectively) are fixed, there is an optimalopening strength at which the coherent transport enhancement is optimized. In Secs. IV and V, respectively, weextend our analysis to two paradigmatic models of transport: the linear chain and the fully connected network. Thelinear chain in particular has been widely considered in the literature [7, 18, 33, 34], and the fully connected networkhas been explored in Ref. [34]. Finally, in Sec. VI we consider the Fenna-Matthews-Olson (FMO) light-harvestingcomplex, and demonstrate that the opening-assisted coherent transport obtained analytically in the earlier models isalso present in this naturally occurring system. In Sec. VII we present our conclusions. II. QUANTUM NETWORKS
Here we present the quantum network models that we will consider. A quantum network is a tight binding modelwhere an excitation can hop from site to site in a specified geometry.The first example is the linear chain, see the upper panel of Fig. 1. This model has been widely analyzed in theliterature due to its relevance in natural and artificial energy transport devices, and is characterized by the followingsystem Hamiltonian ( (cid:126) = 1 here and in the following):H lin = N (cid:88) i =1 ω i | i (cid:105) (cid:104) i | + Ω N − (cid:88) j =1 ( | j (cid:105) (cid:104) j + 1 | + | j + 1 (cid:105) (cid:104) j | ) , (1)where ω i are the site energies and Ω is the coupling between neighboring sites. Here, | j (cid:105) represents a state in whichthe excitation is at the site j , when all the other sites are unoccupied. In terms of two-level states, | (cid:105) , | (cid:105) , it canbe written as | j (cid:105) = | (cid:105) | (cid:105) . . . | (cid:105) j . . . | (cid:105) N . It is common to introduce static noise by letting the energies ω i fluctuaterandomly in the interval [ − W/ , W/
2] with a uniform distribution, and variance σ = W / H eff ) jk = (H sys ) jk − i (cid:88) c A cj ( A ck ) ∗ ≡ (H sys ) jk − i Q jk , (2)where H sys is the closed system Hamiltonian, e.g. H sys = H lin , and A ci are the transition amplitudes from the discretestates i to the continuum channels c . If we consider a single decay channel, c = 1, coupled to site N with decay rateΓ trap , we have A N = (cid:112) Γ trap /
2, and Q jk = Γ trap δ jN δ kN . Including fluorescence effects, where the excitation may belost from any site with rate Γ fl , we have Q jk = (Γ trap δ jN + Γ fl ) δ jk .The quantum evolution (given by the operator U = e − iH eff t ) is non-unitary, since there is a loss of probabilitydue to the decay channel and fluorescence. The complex eigenvalues of H eff can be written as E r − i Γ r /
2, where Γ r represent the decay widths of the resonances. Superradiance, as discussed in the literature [21, 35], is usually reachedonly above a critical coupling strength with the continuum (in the overlapping resonance regime): (cid:104) Γ (cid:105) /D ≥ , (3)where (cid:104) Γ (cid:105) is the average decay width and D is the mean level spacing of the closed system described by H sys .As a further effect of the environment we consider the dephasing caused by dynamic disorder. To include dephasing,we need to switch to a master equation for the reduced density matrix ρ [36],˙ ρ ( t ) = −L tot ρ ( t ) , (4)where the Liouville superoperator is given by L tot = L sys + L trap + L fl + L deph and the four terms respectively describethe dynamics of the closed system, L sys ρ = i [H sys , ρ ] , (5)exciton trapping to the reaction center, L trap ρ = Γ trap {| N (cid:105) (cid:104) N | , ρ } , (6)decay due to fluorescence, L fl ρ = Γ fl ρ , (7)and the dephasing effect as described in the simplest approximation by the Haken-Strobl-Reineker (HSR) model [36]with dephasing rate γ , ( L deph ρ ) jk = γρ jk (1 − δ jk ) . (8)The efficiency of exciton transport can be measured by the total population trapped by the sink [18, 34], η = Γ trap (cid:90) ∞ ρ NN ( t ) dt , (9)or by the average transfer time to reach the sink [17], τ = Γ trap η (cid:90) ∞ t ρ NN ( t ) dt . (10)The system is initiated with one exciton at site 1, i.e., ρ (0) = | (cid:105) (cid:104) | . Formally the solutions for η and τ can bewritten as, η = Γ trap ( L − ρ (0)) NN (11)and τ = Γ trap η ( L − ρ (0)) NN . (12)In physical applications, we are typically interested in the parameter regime of high efficiency η , which can occur onlywhen fluorescence is weak, i.e., when the fluorescence rate Γ fl is smaller than both the trapping rate Γ trap and theenergy scales in the closed-system Hamiltonian H sys . The FMO complex discussed in Sec. VI is a typical example:here the exciton recombination time 1 / Γ fl is estimated to be around 1 ns, whereas the other times scales in theproblem are of the order of picoseconds or tens of picoseconds [17, 18, 34]. In this regime, the effect of Γ fl on theefficiency η and transfer time τ may be treated perturbatively (see e.g. Ref. [37]): Specifically, τ is independent of Γ fl to leading order, and η is related to τ by η = 11 + Γ fl τ (13)when higher-order corrections are omitted. Thus, for a given fluorescence rate, maximizing efficiency η is entirelyequivalent to minimizing the transfer time τ . In the following, we will assume for simplicity of presentation that Γ fl is indeed small, and will present results for τ only; analogous expressions for the efficiency η may be easily obtainedby inserting these results into Eq. (13).In the following, we will be interested in the disorder-ensemble averaged transfer time, defined as (cid:104) τ (cid:105) W = 1 W N (cid:90) W/ − W/ .. (cid:90) W/ − W/ τ ( ω , ω , ..ω N ) dω dω . . . dω N . (14) III. TWO-SITE MODELA. F¨orster approximation
In the 1940s, F¨orster [38] proposed an incoherent non-radiative resonance theory of the energy transfer process inweakly coupled pigments. This mechanism was based on the assumption that, due to large dephasing, the motionof an excitation between chromophores is a classical random walk, which can be described by an incoherent masterequation.Let us first consider a dimer of interacting chromophores and the transmission of the excitation from one moleculeto the other. The Hamiltonian of the system is H = (cid:18) ω ΩΩ ω (cid:19) , (15)where Ω and ω − ω = ∆ are respectively the coupling and the excitation energy difference between the two molecules.Note that | (cid:105) represents a state where molecule 1 is excited and molecule 2 is in its ground state.The energy difference or detuning ∆ is entirely due to the interaction with the environment, if we assume themolecules of the dimer to be identical. The exciton-coupled dimer is most productively viewed as a supermoleculewith two delocalized electronic transitions, rather than a pair of individual molecules, which means switching to thebasis that diagonalizes H .For this Hamiltonian, the probability for an initial excitation in the first molecule to move to the second one isgiven by P → ( t ) = 4Ω + ∆ sin (cid:16)(cid:112) + ∆ t/ (cid:17) , (16)to which we can associate a typical hopping time τ hop = π/ √ + ∆ , a very important parameter for understandingthe propagation.In the F¨orster theory, dephasing is assumed to be large. If γ (cid:29) /τ hop , the dephasing time is much smaller thanthe hopping time, τ d = 1 /γ (cid:28) τ hop . In this regime, coherence is suppressed and exciton dynamics becomes diffusive.The transfer rate from one molecule to the other is given by: T → ∼ dP → ( τ d ) dτ d ≈ γ . (17)This transfer rate also gives the diffusion coefficient for a linear chain of chromophores coupled by a nearest-neighborinteraction, as considered in Refs. [7, 19] for Ω (cid:29) ∆. Indeed, the mean squared number of steps that an excitation canmove is proportional to the time measured in units of the average transfer time τ = 1 /T → , i.e., r ( t ) ∝ t/τ = T → t .The diffusion coefficient in this regime is thus given by Eq. (17) and it agrees with previous results [7, 19] in the sameregime.If dephasing is still large compared to the coupling Ω, but small compared to the detuning ∆, ∆ (cid:29) γ (cid:29) Ω , wemust average P → ( t ) over time and obtain T → = P → τ d ≈ γ ∆ (18)This expression also agrees with the diffusion coefficient given in [7, 19] in the same regime.In general, as long as γ (cid:29) Ω holds, we have the F¨orster transition rate T F = 2Ω γγ + ∆ , (19)with the scalings given by Eqs. (17) and (18) as special cases.Here we will not discuss the weak dephasing regime γ < Ω in which F¨orster theory does not apply. This regimehas been investigated in [7, 19], where it was shown that the excitation dynamics is still diffusive, but with mean freepath of order the localization length, so that the diffusion coefficient is enhanced by the localization length squared.
B. Two-site model with opening
The same two-site system can be considered in the most general context in which the interaction with a sink isexplicitly taken into account. For this purpose we add to the two-site Hamiltonian described in the previous sectiona term representing the possibility of escaping from state | (cid:105) to an external continuum with decay rate Γ trap , seeFig. 2 (left panel). Moreover the system is in contact with another environment that induces fast time-dependentfluctuations of the site energies with variance proportional to γ . The presence of detuning strongly suppresses theprobability of the excitation leaving the system. On the other hand, dephasing produces an energy broadening,which facilitates transport. For very large dephasing, the probability for the two site energies to match becomessmall and thus transport is again suppressed. Optimal transport thus occurs at some intermediate dephasing value: γ ≈ ∆ [17, 18]. This is the noise assisted transport: Noise can help in a situation where transport is suppressed inpresence of only coherent motion.Another general principle that is essential for understanding transport efficiency in open systems is superradiance.Indeed, due to the coupling with a continuum of states, the state | (cid:105) has an energy broadening Γ trap , even in absenceof dephasing, which can also facilitate transport. The system in the absence of dephasing is described by the following2 × H eff = (cid:18) ω ΩΩ ω − i Γ trap / (cid:19) . (20)The complex eigenvalues (taking ω = 0 and ω = ∆) are: E ± = ∆2 − i Γ trap ± (cid:114) (∆ − i Γ trap + 4Ω , (21)and their imaginary parts represent the decay widths of the system. As a function of Γ trap , one of the decay widthshas a non-monotonic behavior which signals the superradiance transition (ST), see Fig. 2 (upper right panel). For∆ (cid:29) Ω, this transition, corresponding to the maximum of the smaller width, occurs at Γ trap ≈ trap , the transport becomes more efficientwith increasing Γ trap , since the decay width of both states increases. On the other side, above the ST, only one ofthe two decay widths continues to increase with Γ trap , while the other decreases. At the same time, the state withthe larger decay width becomes localized on site | (cid:105) , thus suppressing transport.Note that while noise-assisted transport occurs only in presence of a detuning ∆, superradiance-assisted transport(SAT) occurs even with ∆ = 0 and in the absence of dephasing.The non-monotonic behavior of the transfer time as a function of Γ trap is a purely quantum coherent effect. To seethis effect analytically, we consider the master equation (4), which for the two-site model can be written explicitly as ˙ ρ ˙ ρ ˙ ρ ˙ ρ = i Ω − i Ω 0 i Ω i ∆ − Γ trap − γ − i Ω − i Ω 0 − i ∆ − Γ trap − γ i Ω0 − i Ω i Ω − Γ trap ρ ρ ρ ρ . (22) -2 -1 G trap -4 -2 -I m ( E ) -2 -1 G trap t QuantumFörster -i G trap /2 WDg g |1> |2> ST FIG. 2: (Color online) Left panel: Schematic view of the two-site model in the presence of dephasing and coupling to thesink. Upper right panel: The imaginary part of the eigenvalues of the non-Hermitian Hamiltonian given in Eq. (20). HereΩ = 1 , ∆ = 10 , γ = 0. Lower right panel: Transfer time as a function of decay width to the sink for the quantum model (solidcurve) and for the F¨orster model (dashed curve). The vertical dashed line represents the superradiance transition (ST). HereΩ = 0 . γ = 1 , ∆ = 10. Following [37] we may insert the stationary solution ( ˙ ρ = ˙ ρ = 0) for the off-diagonal matrix elements into Eq. (22)and obtain a rate equation for the populations ρ and ρ only: (cid:18) ˙ ρ ˙ ρ (cid:19) = (cid:18) − T → T → T → − T → − Γ trap (cid:19) (cid:18) ρ ρ (cid:19) . (23)These transition rates have been derived by Leegwater in [39]. In our case we have T → = T → = T L with T L = 2Ω ( γ + Γ trap / γ + Γ trap / + ∆ . (24)The incoherent master equation given in Eq. (23) represents a good approximation of the exact quantum dynamics,Eq. (22), when the off-diagonal matrix elements reach a stationary solution very fast. This is valid when the dephasingis sufficiently fast: γ (cid:29) Ω , (25)which is the same condition as the one that ensures validity of the F¨orster transition rate approximation (19) in theclosed system. We observe that the Leegwater rate given by Eq. (25) reduces to the F¨orster rate given by Eq. (19) inthe limit where the system is closed, Γ trap → trap . So thetransition probability given in Eq. (24) includes also coherent effects due to the opening. This point of view is slightlydifferent from the one in [37] where the master equation Eq. (23) is viewed as “classical.” From now on we will referto the master equation (22) with transition rates given in Eq. (24) as the Leegwater model, while the F¨orster modelwill denote the master equation (22) with the F¨orster transition rates (19), independent of Γ trap . Needless to say, T L (Γ trap = 0) = T F .Now two questions present themselves. First, we would like to understand which values of the opening strengthΓ trap cause the F¨orster model to fail due to the coherent effects induced by the opening. Comparing Eqs. (19) and(24), it is clear that the F¨orster model applies when Eq. (25) holds and Γ trap / (cid:28) γ . Even in the presence of largedephasing, when Γ trap is also large (and becomes of the order of γ ), coherent effects cannot be neglected and quantumtransport differs significantly from that predicted by the F¨orster theory (compare the red dashed curve with the solidblack curve in Fig. 2 (lower right panel).Second, we would like to address whether quantum effects can provide enhancement over the transport predictedby F¨orster theory. A clear example showing that this can happen appears in Fig. 2 (lower right panel), where, for alarge region of values of Γ trap , the quantum transfer time is significantly less than that predicted by F¨orster theory.So the idea is the following: Even in presence of large dephasing, for which a F¨orster model of incoherent transport isexpected to apply, as we increase the coupling Γ trap to a sink, coherent effects can be revived and enhance transport.Finding overall conditions for optimal transport in open quantum systems will be a key focus of the following analysis.Below we will derive analytical expressions for the transfer times and address the above questions quantitatively. C. Transfer time, optimal opening, and quantum enhancement
In the two-site case, one can obtain a simple yet exact analytic form for the transport time τ , Eq. (10), usingEq. (12) and substituting the exact Liouville operator L given by Eq. (22) [37]: τ = 12Ω (cid:32) Γ trap + γ + Γ trap γ + Γ trap (cid:33) . (26)Eq. (26) shows explicitly the non-monotonic behavior of the transfer time with the opening Γ trap , which is a signatureof quantum coherence and is clearly visible in Fig. 2 (lower right panel).The expression for the average transfer time can be aso computed using the incoherent master equation (22), witheither the F¨orster or Leegwater transition rate. While for the two-site case the Leegwater average transfer time isexactly the same as the full quantum result (26), for the F¨orster theory we have: τ F = 12Ω (cid:18) Γ trap + γ + ∆ γ (cid:19) . (27)We note that τ F decays monotonically with increasing opening Γ trap , as it must in a classical calculation. Clearly forΓ trap (cid:28) γ , F¨orster theory coincides with the full quantum result. On the other hand for Γ trap (cid:38) γ , coherent effectsbecome important and they can be incorporated using the Leegwater model (at least for the two-site case).Since τ F is a monotonic function of Γ trap , it assumes its minimum value τ minF = 12Ω (cid:18) γ + ∆ γ (cid:19) (28)for Γ trap → ∞ . On the other hand, the quantum transfer time is minimized at a finite value of Γ trap . Unfortunatelythe optimal value of Γ trap is given in general by the solution to a quartic equation. Nevertheless it is easy to obtainsimple expressions in several physically relevant regimes. In particular, of greatest physical interest is the situationwhere the quantum minimum associated with optimal value of the opening is deep, which is only possible where alarge difference exists in the first place between quantum and incoherent transport, i.e., Γ trap (cid:29) γ , as discussed abovein Sec. III. In that regime, the optimal opening is given byΓ opttrap ≈ (cid:112) ∆ + 2Ω − γ ∆ ∆ + 2Ω + O ( γ ) . (29)If in addition to dephasing being weak, detuning is strong (∆ (cid:29) Ω), Eq. (29) simplifies toΓ opttrap ≈ − γ . (30)In Fig. 3 (left panel) the simple analytical expression (30) is shown to agree very well with exact numerical calculationsfor the quantum model. This result is particularly interesting since it shows the effect of dephasing on the ST: Whilefor small dephasing the optimal opening strength is given by the ST criterion Γ ST ≈ trap = Γ ST − γ decreases with the dephasing γ .The condition for optimal transport given in Eq. (30), can be re-written as ∆ = γ +Γ opttrap /
2. This can be interpretedby saying that dephasing and opening together induce a cumulative energy broadening, which optimize transport whenit matches the detuning ∆. Also striking is the symmetrical role that γ and Γ trap play in controlling transport efficiencyeven if their origin and underlying physics are completely different. For instance γ induces dephasing in the system,whereas Γ trap increases the coherent effects.The optimal dephasing, fixing all other variables, is given exactly by γ opt = ∆ − Γ trap / , (31) -2 -1 g/D t m i n FörsterQuantum10 -4 -3 -2 -1 g/D G t r a pop t Quantumtheor
D>> g, W ST a) b) FIG. 3: (Color online) Left panel: Optimal coupling to the sink, Γ opttrap , in a two-site system, as a function of the rescaleddephasing strength γ/ ∆ in the regime ∆ (cid:29) Ω. Data refer to the case ∆ = 100 , Ω = 1. Symbols represent numerical simulationsof the full quantum model, the dashed red curve shows the analytical result given by Eq. (30), and the blue arrow shows theasymptotic value given by Eq. (34). The solid horizontal line indicates the value at which the superradiance transition (ST)occurs for zero dephasing. Right panel: Minimal transfer times for the F¨orster model (solid curve) and for the full quantumcalculation (symbols) are shown as functions of the rescaled dephasing strength γ/ ∆. in any regime. This shows that also the criterion for noise assisted transport, γ ≈ ∆, is modified by the presence of astrong opening.For the value of Γ trap given in Eq. (30), the minimal transfer time assumes the value: τ min ≈ ∆Ω , (32)which should be compared with the F¨orster expression (28) in the same regime, τ minF ≈ ∆ γ Ω = τ min ∆2 γ (cid:29) τ min , (33)showing that in this regime (∆ (cid:29) γ (cid:29) Ω) quantum coherence, induced by the coupling to the sink, can alwaysenhance transport.In the opposite limit Ω (cid:29) ∆ (cid:29) γ one obtains: Γ opttrap = 2 √ . (34)We summarize our results so far in the following way: For very small opening, Γ trap (cid:28) γ , the F¨orster modelreproduces the quantum results. In the opposite limit Γ trap → ∞ , quantum transport is always fully suppressed whilethe F¨orster model prediction for the average transfer time approaches a non-zero asymptotic value, thus showing thenon-applicability of this model. In general, we expect the F¨orster model to fail when Γ trap (cid:38) γ , so that coherenteffects that occur on a time scale 1 / Γ trap , can be relevant before dephasing destroys them on a time scale 1 /γ .What is the regime in which quantum transport is better than the incoherent transport described by the F¨orstermodel? In order to find this regime, let us write the difference between the two transfer times: τ F − τ = Γ trap (cid:20) ∆ γ ( γ + Γ trap / − (cid:21) . (35)Clearly, quantum transfer is enhanced over the F¨orster prediction ( τ < τ F ) if and only if γ + Γ trap / < ∆ /γ . Onthe other hand, as noted earlier, the relative difference is small, i.e., the F¨orster model is a good approximation, whenΓ trap / (cid:28) γ . So the regime where quantum coherent effects produce a significant enhancement of transfer efficiencyin the two-site model is given by γ (cid:46) Γ trap < ∆ γ − γ . (36) . . . . γ/Δ -2 -1 Γ t r a p / Δ F ö rster regimeQuantum enhancementQuantum Zeno FIG. 4: (Color online) The parameter regime of significant quantum enhancement of transport in the two-site model, Eq. (36),and the F¨orster regime, γ < Γ trap /
2, where quantum effects are negligible. In the third regime, corresponding to very largeopening Γ trap , quantum mechanics suppresses transport due to the quantum Zeno effect. Generalization to a disordered linearchain of arbitrary length is obtained by replacing the detuning ∆ in the two-site model with W/ √
6, where W is the disorderstrength. This result is consistent with the illustration in Fig. 2 (lower right panel): when the opening is very small, the F¨orsterapproximation holds, whereas for very large opening, coherent effects cause trapping (a quantum Zeno effect). It isonly for the range of openings given in Eq. (36) that quantum coherence aids transport.The quantum transport enhancement regime given by Eq.( 36) is illustrated graphically in Fig. 4. As the dephasingdecreases, quantum enhancement of transport occurs for an ever wider range of openings Γ trap . We also observethat near the superradiance transition, Γ trap ∼ γ .Note that in the case of static disorder, the disorder-averaged transfer time can be computed as stated in Eq. (14).The results of this section remain valid if we substitute ∆ with (cid:104) ( ω − ω ) (cid:105) = W / IV. LONG CHAINS WITH STATIC DISORDERA. Linear chain: Analytic results
For a linear chain with N sites, see Fig. 1 (upper panel), in the presence of static disorder, it is not possible toget an analytical expression for the full quantum model. Nevertheless under the strong dephasing condition given inEq. (25) the dynamics of the system can be described by the incoherent master equation, dP j dt = (cid:88) k ( T k → j P k − T j → k P j ) − δ j,n Γ trap P j , (37)where P j is the probability to be at site j . The nearest-neighbor transfer rates in Eq. (37), T k → j , are given by T F ,Eq. (19), with the exception of the transfer rate along the bond adjacent to the sink, T n − → n = T n → n − , which isgiven by the Leegwater expression T L , Eq. (24). The F¨orster model is also given by Eq. (37) but with all the transferrates given by T F , Eq. (19).Proceeding in the same way as for the case N = 2, we obtain analytical expressions for the ensemble-averagedF¨orster and Leegwater transfer times: (cid:104) τ F (cid:105) W = N Γ trap + N ( N − (cid:18) γ + W γ (cid:19) (38)0and (cid:104) τ L (cid:105) W = N Γ trap + N ( N − (cid:20) γ + Γ trap N + W γ (cid:18) − trap N (2 γ + Γ trap ) (cid:19)(cid:21) . (39)The effect of quantum coherence is given by the difference in transfer times, (cid:104) τ F (cid:105) W − (cid:104) τ L (cid:105) W = ( N − trap (cid:18) W γ (2 γ + Γ trap ) − (cid:19) . (40)In general, increasing the ratio W/γ (i.e., increasing the strength of static as compared to dynamical disorder) willmake the difference in Eq. (40) more positive, i.e., quantum transport becomes more favored relative to incoherenttransport, just as has been seen already in the two-site case. To be precise, from Eq. (40), W γ (2 γ + Γ trap ) > ⇒ W > √ γ must hold in order to have (cid:104) τ F (cid:105) W > (cid:104) τ L (cid:105) W , and the regime where quantum effects are both helpful and significantis then identical to the one identified in Eq. (36) and Fig. 4 for the two-site model, with the simple replacement∆ → W / γ (cid:46) Γ trap < W γ − γ . (41)We notice that W > √ γ is a necessary condition for significant quantum transport enhancement to occur, i.e., staticdisorder must be stronger than dynamic disorder. Given W > √ γ , (cid:104) τ F (cid:105) W − (cid:104) τ L (cid:105) W is maximized when Γ trap = Γ opttrap ,where Γ opttrap = 2 (cid:16) W/ √ − γ (cid:17) . (42)To be precise, we should note that the value of Γ trap that maximizes the transport enhancement (cid:104) τ F (cid:105) W − (cid:104) τ L (cid:105) W is notexactly the same as the value at which the quantum transport time (cid:104) τ L (cid:105) W is minimized, but in the limit of small Ωwhere the F¨orster approximation is meaningful, the difference is negligible. In the limiting case W ∼ Γ trap (cid:29) γ (cid:29) Ωand N (cid:29) (cid:104) τ F (cid:105) W ≈ (cid:104) τ L (cid:105) W ≈ N W / γ Ω , so both types of transport are diffusive, while the difference in transfertimes (cid:104) τ F (cid:105) W − (cid:104) τ L (cid:105) W is N W / γ Ω . In relative terms, quantum enhancement is therefore most important in shortchains, which is a case relevant in realistic photosynthetic complexes where the number of chromophores is small, e.g.the FMO complex which will be the focus of Sec. VI. B. Linear chain: Numerical results
Eqs. (38) and (39) provide, respectively, the analytical results for the average transfer time in the F¨orster model,which is purely incoherent, and in the Leegwater approximation, which incorporates quantum coherence effects.Unfortunately no analytic result is available for the exact quantum calculation in a chain of general length N and forthis reason we will present results obtained by means of numerical simulations. In particular we will show that: • The Leegwater expression, Eq. (39), provides a good approximation for the exact quantum transfer time in theregime of interest given by Eq. (41), as long as the semiclassical condition, γ (cid:29) Ω, holds; • Equation (42), obtained from the analytic Leegwater calculation, accurately describes the opening at which theexact quantum transport enhancement (cid:104) τ F (cid:105) W − (cid:104) τ (cid:105) W is maximized; • Quantum corrections beyond the Leegwater approximation give rise to even stronger quantum transport en-hancement when the semiclassical condition γ (cid:29) Ω is no longer satisfied.First, results for N = 3 sites are reported in Fig. 5. We see that in the case W (cid:29) γ (cid:29) Ω ( W = 100, γ = 10), theexact quantum results agree very well with the Leegwater approximation, and the maximal quantum enhancementoccurs at Γ trap ≈ Γ opttrap , as predicted. For W ∼ γ (see W = 3 , γ = 1), quantum transport enhancement becomesnegligible, and the enhancement effect disappears entirely for W (cid:46) γ (see W = 0 . , γ = 0 . -2 -1 Γ trap / Γ trap -1 < τ > W W=10, γ=1
W=100, γ=10
W=3 γ=1
W=0.3, γ=0.3 opt
FIG. 5: (Color online) F¨orster (solid), Leegwater (dotted), and quantum (symbols) average transfer times for the N = 3 linearchain. Here we fix Ω = 1. On the horizontal axis we normalize the trapping by Γ opttrap as given by Eq. (42). than that predicted by the Leegwater model (see for example W = 10 , γ = 1 in Fig. 5). This correction is addressedat a quantitative level below.Next, we confirm that these results continue to hold for long chains ( N (cid:29) N = 10and 20 a fixed set of parameters such that W (cid:29) γ (cid:29) Ω. It is clear from Fig. 6 that the above picture continues tohold at large N : The difference between incoherent and quantum transfer times is still maximized for Γ trap (cid:39) Γ opttrap ,the minimal quantum transfer time also occurs near Γ opttrap , and the Leegwater approximation underpredicts the truequantum enhancement by a slight margin.We now briefly return to the observation in Figs. 5 and 6 of a slight discrepancy between the exact quantumcalculation and the Leegwater expressions. Although no analytic expression is available for the quantum chain with N >
2, a systematic numerical scaling analysis in the regime of interest, N (cid:29) W (cid:29) γ (cid:29) Ω, shows that the leadingquantum correction to the Leegwater formula scales as (cid:104) τ (cid:105) W − (cid:104) τ L (cid:105) W = − bN W/γ , (43)where b ≈ .
72 is a constant, see Fig. 7. Comparing with Eq. (39), we find the relative error in the Leegwaterapproximation: (cid:104) τ (cid:105) W / (cid:104) τ L (cid:105) W = 1 − O (Ω /W γ ) , (44)independent of chain length for N (cid:29)
1. This agrees with our previous observations: the Leegwater approach providesan excellent approximation to the quantum transfer time in the regime where quantum enhancement is possible, andthe leading correction favors even slightly faster transport than that predicted by the Leegwater formula.
C. Heuristic derivation of transfer times for the linear chain
Here we give an heuristic derivation of the average transfer time obtained in the previous section. In particular wewill analyze the parameter regime where quantum transport outperforms incoherent transport.Consider a linear chain of N sites. We start at one end of the chain and evaluate the probability to reach the otherend where the excitation can escape with a rate Γ trap .Let us first compute the average transfer time in the F¨orster model. To go from site 1 to site 2 takes an averagetime 1 /T F . The total time required to perform the random walk from site 1 to site N scales as N , or more precisely,2 -2 -1 Γ trap / Γ trap < τ > W / N N=10 Quantum N=10 FörsterN=10 LeegwaterN=20 QuantumN=20 FörsterN=20 Leegwater opt
FIG. 6: (Color online) Comparison among F¨orster (solid cures), Leegwater (dotted and dashed curves), and quantum (symbols)average transfer times (rescaled by a factor N ) for a linear chain with N = 10 and N = 20 sites. Here we fix Ω = 1, γ = 10,and W = 50. On the horizontal axis we renormalize the trapping by Γ opttrap (Eq. (42)).am N
10 20 30 40 50 = " = W=1000, ! trap =800, . =10 N
10 15 20 25 30 35 40 p ! " = W=1000, ! trap =800, . =10 .
10 15 20 25 30 35 40 45 p " = N=10, W=5000, ! trap =4000 .
10 15 20 25 30 35 40 45 = p ! " = N=10, W=1000, ! trap =800 W = " = N=10, ! trap =0.8W, . =10 W
100 200 300 400 500 ! " = N=10, ! trap =0.8W, . =5 FIG. 7: (Color online) The error in the Leegwater approximation, ∆ τ = (cid:104) τ (cid:105) − (cid:104) τ L (cid:105) , is shown as a function of system size N ,dephasing strength γ , and disorder strength W , in the parameter regime of interest. The panels on the left side are for thefully-connected network and those on the right side are for the linear chain. The solid blue lines in the two top panels arefits to ( a (cid:48) + b (cid:48) N ) / √ γW for the fully-connected network (with a (cid:48) = 2 . b (cid:48) = 0 .
37) and to − ( a + bN ) W/γ for the chain(with a = 1 . b = 0 . N limit. We fix Ω = 1, and theother parameters are shown in the legend of each panel. Specially we fix Γ trap = 0 . W , which is close to the optimal openingΓ opttrap = √ W/ N ( N − /T F . Moreover, if the probability to be at the N -th site is 1 /N and the escape rate is Γ trap we can estimatethe exit time as N/ Γ trap . Adding up the diffusion time and the exit time we have, τ F = N ( N − T F + N Γ trap = N Γ trap + N ( N − (cid:18) γ + W γ (cid:19) , (45)which is exactly the result found by direct calculation, see Eq. (38). On the other hand, in the presence of an opening,the Leegwater formulas are modified by the substitution γ → γ + Γ trap / N − N -th site where the excitation can escape). Needlessto say, while for Γ trap (cid:28) γ the transfer rate in presence of the opening reduces to the incoherent one, for Γ trap ≈ γ the two rates are very different. In particular the rate is maximal for γ + Γ trap / W/ √ τ L = N Γ trap + ( N − N − T F + N − T L . (46)This expression, rearranged, is the same as Eq. (39). From the above expression we get, τ F − τ L = ( N − (cid:18) T F − T L (cid:19) . (47)This last expression is simpler to analyze: quantum transport is better than incoherent transport when T L > T F , fromwhich we have: Γ trap < W − γ γ (48)which can be achieved only if W > √ γ . Moreover for W (cid:29) Ω and Γ trap > γ , the optimal quantum transport isobtained for Γ trap = Γ opttrap ≡ (cid:112) / W − γ (the same value that maximizes the rate T L ). We can now compare theoptimal quantum transport with the optimal F¨orster transport obtained for Γ trap → ∞ . So we have: τ optF = N ( N − T F , (49) τ optL = ( N − N − T F + N − T optL + N Γ opttrap (50)and for W (cid:29) γ , N (cid:29) τ optF − τ optL ≈ N W γ . (51)Note that that quantum enhancement due to the opening is proportional to the variance of the static disorder. V. FULLY CONNECTED NETWORKS
In Sec. IV we saw that coherent effects can aid transport through an open linear chain of arbitrary length, as longas the static disorder is sufficiently strong relative to the dephasing rate. To demonstrate the generality of this effect,we now consider a quantum network which, in its degree of connectivity, may be considered to be at the oppositeextreme from a linear chain, namely a fully connected network with equal couplings between all pairs of sites, asillustrated in Fig. 1 (lower panel). Specifically, the Hamiltonian H lin (Eq. (1)) for the linear chain is replaced byH fc = N (cid:88) i =1 ω i | i (cid:105) (cid:104) i | + Ω (cid:88) ≤ i 100 150 200 250 300 C FIG. 8: (Color online) The function C ( N ) in Eq. (60), which describes the N -dependence of the incoherent transfer time inthe fully connected model, with a fit to C ( N ) = 2 . N + 64 . 55. Here W = 5000, Γ trap = 6000, and γ = Ω = 1. transfer rates T j → N and T N → j are given by the modified rate T L (Eq. (24)), which includes the effect of the opening.To simplify the analysis, we focus only on the regime where quantum transport enhancement is most pronounced: Thisoccurs when the opening is comparable to the disorder strength, and the mean level spacing W/N is large comparedto the dephasing rate, Γ trap /N ∼ W/N (cid:29) γ (cid:29) Ω. In that case we have T F ∼ Ω γ/W and T L ∼ Ω /W , so T L (cid:29) T F and to leading order all sites are effectively coupled to site N only. In this regime, the average time to reach the sinkstarting from site 1 attains the N -independent value (cid:104) τ L (cid:105) W = 3Γ + 2 W Γ trap . (53)We note that this expression agrees, as it must, with the linear chain result (39) for the case N = 2 in the limitΓ trap ∼ W (cid:29) γ (cid:29) Ω. The transfer time is minimized, (cid:104) τ L (cid:105) min W = W/ √ , (54)when the opening strength is set to the optimal valueΓ opttrap = (cid:112) / W . (55)The above discussion addresses the transfer time in the context of the Leegwater approximation. As in the caseof the linear chain, an exact numerical evaluation of the average quantum transfer time confirms that the Leegwaterapproximation provides the leading contribution to the quantum transfer time in the regime of interest. The leadingcorrection for the error takes the form (cid:104) τ (cid:105) W − (cid:104) τ L (cid:105) W ≈ ( b (cid:48) N ) / (cid:112) γW , (56)with b (cid:48) ≈ . 37, as illustrated in Fig. 7, so (cid:104) τ (cid:105) W / (cid:104) τ L (cid:105) W = 1 + O ( N Ω /W / γ / ) . (57)To obtain the corresponding F¨orster behavior, it is convenient to work in the large- N limit. The probability tojump from site i to site j is given by the F¨orster transition rate (19),( T F ) i → j = 2Ω γγ + ( ω i − ω j ) . (58)5 ! trap10 -2 -1 h = i W W=500W=500W=500W=50W=50W=50W=5W=5W=5 FIG. 9: (Color online) Comparison among F¨orster (solid curves with symbols), Leegwater (dashed cruves), and quantum(symbols only) average transfer times for a fully connected network with N = 10 sites and several values of the disorderstrength W . Here we fix Ω = 1 and γ = 5. Now if we label the sites in order of site energy, ω < ω < . . . < ω N , for large N we have ( ω i − ω j ) ≈ W ( i − j ) /N . Since we are working in the regime of very strong disorder, W (cid:29) N γ , the transition rates simplify to( T F ) i → j ≈ N γ Ω /W ( i − j ) . This corresponds to an α = 1 L´evy flight (or Cauchy flight) with typical time scale∆ t ∼ W /N γ Ω for each jump; for an α = 1 L´evy flight the average time to travel a distance n scales as n , incontrast with the n scaling of the travel time for ordinary diffusion [40]. Although the initial site is not necessarilysite 1 due to the site relabeling, and the site coupled to the sink is not necessarily site N , the initial and final sitesare nevertheless separated by a distance of order N . Thus, total time required to travel through the system scales as τ ∼ N ∆ t , and we have (cid:104) τ F (cid:105) W ∼ W N γ Ω . (59)The behavior given in Eq. (59) is confirmed by exact numerical calculations. Numerically we obtain an excellent fitto (cid:104) τ F (cid:105) W ≈ W C ( N ) γ Ω , (60)where C ( N ) ≈ . N + 64 . 55. We note that from Eq. (59) or Eq. (60) the incoherent transfer time may appear toapproach 0 in the large- N limit; however one must keep in mind that the above discussion assumes W/N (cid:29) γ . If weincrease N while holding all other system parameters fixed, we find instead that for N > W/γ , the F¨orster transfertime saturates at an N -independent value (cid:104) τ F (cid:105) W ∼ W/ Ω , comparable to the Leegwater prediction. In this limitthere is no significant quantum enhancement of transport.Returning to the regime of primary interest, W/N (cid:29) γ (cid:29) Ω and comparing Eqs. (54) and (59) we find a verystrong coherent enhancement of transport in the fully connected network. Specifically, when the opening strengthΓ trap is of order Γ opttrap , the ratio of the Leegwater (or, equivalently, quantum) transfer time to the incoherent timescales as (cid:104) τ L (cid:105) W (cid:104) τ F (cid:105) W ∼ γW/N (cid:28) . (61)We note that the condition W/N (cid:29) γ which allows quantum mechanics to significantly aid transport in the fullyconnected network corresponds precisely to the starting assumption underlying the calculations in this Section.6 ! trap -2 -1 h = i W N=5N=5N=5N=10N=10N=10N=15N=15N=15 W + ! trap ! trap FIG. 10: (Color online) Comparison among F¨orster (solid curves with symbols), Leegwater (dashed curves), and quantum(symbols only) average transfer times for fully connected networks of several sizes N . Here we fix W = 500, γ = 5, and Ω = 1.The analytic result (53) describes the behavior of the Leegwater and quantum ensemble-averaged transfer time in the regionof strongest quantum transport enhancement. The results in Fig. 9 confirm that a very strong quantum enhancement of transport occurs in a fully connectednetwork of N = 10 sites for W (cid:29) N γ (see the data for W = 500 in the figure). Optimal quantum transport appears atthe value of the opening given by Γ opttrap . We also observe excellent agreement between the exact quantum calculationand the Leegwater approximation in this regime. The Leegwater approximation breaks down for larger relative valuesof the dephasing rate γ (e.g. W = γ = 5), but in this range of parameters quantum effects hinder rather than aidtransport. Fig. 10 illustrates that in the region of strongest quantum transport enhancement (Γ trap ∼ Γ opttrap ), thequantum behavior is indeed approximately N -independent and is well described by the analytic expression given inEq. (53). VI. THE FMO COMPLEX The FMO photosynthetic complex has received a lot of attention in recent years as an example of a biological systemthat exhibits quantum coherence effects even at room temperature [1–3]. In particular, the interplay of opening andnoise in the FMO complex has been already analyzed in Ref. [20], where it was shown that even at room temperature,the superradiance transition is able to enhance transport. Here we examine opening-assisted quantum transportenhancement in the FMO complex and observe that the same behavior obtains here as in the linear chain and fullyconnected model systems considered in the previous sections.Each subunit of the FMO complex contains seven chromophores, and may be modeled by the tight-binding Hamil-7 Exciton Initialization RC ( Sink ) FIG. 11: (Color online) A schematic illustration of the FMO Hamiltonian (62), with each bond indicating a coupling matrixelement of magnitude at least 20 cm − . tonian H FMO = − . . − . . − . − . − . . . . . . . . − . − . − . − . . − . − . − − . . . − . − . . − . − . . − . − 17 81 . . − . . − . − . . cm − , (62)in units where hc = 1. We notice that the connectivity between the sites is greater than that in a linear chain, but theinter-site couplings are very non-uniform. Thus, this realistic system may be considered to be intermediate betweena chain and a fully connected network. A schematic illustration of the Hamiltonian H FMO , where only off-diagonalelements of magnitude greater than 20 cm − are indicated by bonds, appears in Fig. 11 (but in all calculations belowwe employ the full Hamiltonian given in Eq. (62)).Since incident photons are believed to create excitations on sites 1 and 6 of the FMO complex [17], we take theinitial state of the system to be ρ (0) = 12 ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) . (63)Site 3 is coupled to the reaction center, which serves as the sink for the FMO complex, with decay rate Γ trap .Additionally, an excitation on any site may decay through exciton recombination with rate Γ fl = (1 ns) − =0 . 033 cm − [17, 18], but this slow decay has a negligible effect on the transfer time τ , as discussed in Sec. II.The transfer time calculation as a function of reaction center coupling Γ trap is shown in Fig. 12 (see also Ref. [20]).For the FMO system, the dephasing rate γ is related to the temperature by the relation γ = 0 . c ( T / K) cm − , where T / K is the temperature in Kelvin units [2], and results for three values of the temperature (or equivalently, dephasingrate) are shown in the figure. Notably, strong opening-assisted quantum enhancement of transport is seen not onlyat liquid nitrogen temperature (77 K) but also at room temperature (300 K) where the quantum transfer is up to afactor of 2 faster than that obtained by an incoherent calculation. We also see good agreement between the exactquantum calculation and the Leegwater approximation at room temperature. For comparison, we show an exampleat very high temperature (1500 K), where the quantum transport enhancement is almost absent.Although the “disorder” in the FMO Hamiltonian is fixed, for the purpose of estimating the relevant energy, time,and temperature scales we may analogize this Hamiltonian to one drawn from a disordered ensemble. The varianceof the site energies ( H FMO ) ii is σ = (128 cm − ) , which corresponds to W = 443 cm − . Then we see that roomtemperature, T = 300 K or γ = 156 cm − , actually corresponds to a marginal case where “static disorder” W anddephasing rate γ are comparable. At even higher (biologically unrealistic) temperatures, e.g. T = 1500 K, we have γ = 780 cm − (cid:29) W , and quantum enhancement of transport is absent, as expected. At lower (also unrealistic)8 ! trap (ps -1 ) = ( p s ) T=77KT=77KT=77KT=300KT=300KT=300KT=1500KT=1500KT=1500K FIG. 12: (Color online) The average transfer time for the FMO system is shown as a function of coupling Γ trap between site 3and the reaction center, at three different temperatures. The F¨orster, Leegwater, and quantum transfer times are representedby solid curves with symbols, dashed curves, and symbols, respectively. temperatures, e.g. T = 77 K, we have γ = 40 cm − (cid:28) W , corresponding to a regime where opening-assisted quantumtransport enhancement is most pronounced. The crossover between the low-temperature regime where coherenteffects strongly aid transport and the high-temperature regime where coherent effects provide no advantage is studiedquantitatively in Fig. 13 (upper panel), where the minimal quantum, Leegwater, and F¨orster transfer times are shownat each temperature (optimizing in each case over the opening strength Γ trap ).Similarly we may estimate the optimal strength of the opening at low temperature using the formula Γ opttrap = (cid:112) / W obtained for the linear chain and fully connected network at small dephasing and Ω → opttrap = (cid:112) / πc )(443 cm − ) = 68 ps − , which is in reasonable qualitative agreement with the location of theLeegwater and quantum minima at liquid nitrogen temperature in Fig. 12. (The above formula is valid for inter-sitecouping Ω → 0, and therefore is expected to underestimate the true value of Γ opttrap ). As expected from our study ofthe two-site model and linear chain (see Eqs. (29) and (42)), the location of the minimum shifts to smaller couplingΓ trap as the temperature (dephasing) increases. The full dependence of the optimal opening strength on temperaturein the exact quantum calculation as well as in the Leegwater approximation are shown in detail in Fig. 13 (lowerpanel). VII. CONCLUSIONS We have analyzed the role of the opening in enhancing coherent transport in the presence of both disorder anddephasing. The effect is investigated in several paradigmatic models, including a two-site system, a linear chain of9 -3 -2 -1 = m i n ( p s ) QuantumLeegwaterF B orster Temp (K) -3 -2 -1 ! t r apop t ( p s - ) QuantumLeegwater ST FIG. 13: (Color online) Upper panel: The minimal transfer time through the FMO complex (optimizing over the couplingΓ trap to the reaction center) is shown as a function of temperature, for the quantum, Leegwater, and incoherent (F¨orster)calculations. Lower panel: The optimal coupling Γ opttrap is shown as a function of temperature, in the full quantum calculationand in the Leegwater approximation. In the incoherent model, the optimal coupling is always Γ opttrap = ∞ . The horizontal solidline indicates the location of the superradiance transition at zero temperature. In both panels, the dashed vertical line indicatesroom temperature, T = 300 K. arbitrary length, and a fully connected network of arbitrary size. For the two-site model, fully analytical expressionsexist for both the incoherent and quantum average transfer times, and therefore the regime in which coherent effects aidtransport as well as the optimal opening strength at which the effect is maximized may also be obtained analytically.For the linear chain and fully connected network, we are able to find analytical expressions in the deep classical regime,where dephasing is much stronger than the hopping coupling between the sites. In this case quantum transport canbe described with an incoherent master equation where the rates incorporate the effect of the opening, as suggestedby Leegwater. Again, the different efficiencies of quantum and incoherent transport can be compared to identify theregime in which coherent effects aid transport. In this regime we find the optimal opening able to maximize transportefficiency. We see very generally that quantum transport can outperform incoherent transport even at high rates ofdephasing (or dynamic disorder), as long as the static disorder strength is sufficiently large. The optimal strengthof the opening grows linearly with the disorder strength. 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