Dephasing assisted transport: Quantum networks and biomolecules
aa r X i v : . [ qu a n t - ph ] J u l Dephasing assisted transport: Quantum networks and biomolecules
M. B. Plenio , and S. F. Huelga Institute for Mathematical Sciences, Imperial College London, London SW7 2PG, UK QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK and Quantum Physics Group, Department of Physics, Astronomy & MathematicsUniversity of Hertfordshire, Hatfield, Herts AL10 9AB, UK (Dated: July 30, 2008)Transport phenomena are fundamental in Physics. They allow for information and energy to beexchanged between individual constituents of communication systems, networks or even biologicalentities. Environmental noise will generally hinder the efficiency of the transport process. However,and contrary to intuition, there are situations in classical systems where thermal fluctuations are ac-tually instrumental in assisting transport phenomena. Here we show that, even at zero temperature,transport of excitations across dissipative quantum networks can be enhanced by local dephasingnoise. We explain the underlying physical mechanisms behind this phenomenon, show that entangle-ment does not play a supportive role and propose possible experimental demonstrations in quantumoptics. We argue that Nature may be routinely exploiting this effect and show that the transportof excitations in light harvesting molecules does benefit from such noise assisted processes. Theseresults point towards the possibility for designing optimized structures for transport, for examplein artificial nano-structures, assisted by noise.
Introduction –
Noise is an inevitable feature of anyphysical system, be it natural or artificial. Typically,the presence of noise is associated with the deteriorationof performance for fundamental processes such as infor-mation processing and storage, sensing or transport, insystems ranging from proteins to computing devices.However, the presence of noise does not always hinderthe efficiency of an information process and biologicalsystems provide a paradigm of efficient performance as-sisted by a noisy environment [1]. A vivid illustration ofthe counterintuitive role that noise may play is providedby the phenomenon of stochastic resonance (SR)[2]. Herethermal noise may enhance the response of the system toa weak coherent signal, optimizing the response at an in-termediate noise level [3]. Some experimental evidencesuggests that biological systems employ SR-like strate-gies to enhance transport and sensing [4, 5]. Noise in theform of thermal fluctuations may also lead to directedtransport in ratchets and play a helpful role in Brownianmotors [6, 7, 8]. It seems therefore natural to try anddraw analogies with complex classical networks so thatthe physical mechanisms that underpin their functioningwhen subject to noise can be perhaps mirrored and even-tually used to optimize the performance of complex quan-tum networks. Recently, tentative first steps towards theexploration of the concept of SR in quantum many-bodysystems [9, 10, 11] and quantum communication chan-nels [12, 13, 14] have been undertaken while other studieshave focused into analyzing the persistence of coherenceeffects in biological systems. In particular, detecting thepresence of quantum entanglement, has been the objectof considerable attention [16, 17]. It was noted, how-ever, that even if found, it would be unclear whethersuch entanglement has any functional importance or issimply the unavoidable by-product of coherent quantumdynamics in such systems [18].Here we show that dephasing noise, which leads to
12 3 4 N+1N
FIG. 1: Sites (blue spheres), modeled here as spin-1/2 parti-cles or qubits, are interacting with each other (dashed line)to form a network. The particles may suffer dissipative lossesas well as dephasing. The red arrow indicates an irreversibletransfer of excitations from the network to a sink that acts asa receiver. the destruction of quantum coherence and entanglementas a result of phase randomization, may neverthelessbe an essential resource to enhance the transport ofexcitations when combined with coherent dynamics.Indeed, we show that a dissipative quantum networksubject to dephasing can exhibit an enhanced capacityfor transmission of classical information when seen asa communication channel, even though its quantumcapacity and quantum coherence are diminished bythe presence of noise. It is the constructive interplaybetween dephasing noise and coherent dynamics, ratherthan the presence of entanglement, that is responsiblefor the improved transport of excitations. Recently,this enhancement of quantum transport due to theinterplay between coherence and the environment hasbeen demonstrated and quantified for chromophoriccomplexes (see [19, 20, 21, 22] and Note Added).In addition to the clarifying nature of these results, it isintriguing to observe that Nature appears to exploit noiseassisted processes to maximize the system’s performanceand it will be worthwhile to explore how similar processesmay be useful for the design of improved transport innano-structures and perhaps even quantum informationprocessors.
The basic setting –
We consider a network of N sitesthat may support excitations which can be exchangedbetween lattice sites by hopping (see Fig. 1). The Hamil-tonian that describes this situation is then given by H = N X k =1 ~ ω k σ + k σ − k + X k = l ~ v k,l ( σ − k σ + l + σ + k σ − l ) , (1)where σ + k ( σ − k ) are the raising and lowering operators forsite k , ~ ω k is the local site excitation energies and v k,l denotes the hopping rate of an excitation between thesites k and l . It should be noted that the dynamics inthis system preserves the total excitation number in thesystem. This is not an essential feature but makes thesystem amenable to efficient numerical analysis. We willassume that the system is susceptible simultaneously totwo distinct types of noise processes, a dissipative pro-cess that reduces the number of excitations in the systemat rate Γ k and a dephasing process that randomizes thephase of local excitations at rate γ k .Initially we will assume that we can describe both pro-cesses by using a Markovian master equation with localdephasing and dissipation terms. It is important to notehowever that the effects found here persist when tak-ing account of the system-environment interaction in amore detailed manner (see Methods). Dissipative pro-cesses, which lead to energy loss, are then described bythe Lindblad super-operator L diss ( ρ ) = N X k =1 Γ k [ −{ σ + k σ − k , ρ } + 2 σ − k ρσ + k ] , (2)while energy-conserving dephasing processes are de-scribed by the operator L deph ( ρ ) = N X k =1 γ k [ −{ σ + k σ ( − ) k , ρ } + 2 σ + k σ − k ρσ + k σ − k ] . (3)Finally, in order to be able to measure the total transferof excitation, we designate an additional site, numbered N + 1, which is populated by an irreversible decay pro-cess from a chosen level k as described by the Lindbladoperator L sink ( ρ ) = (4)Γ N +1 [ −{ σ + k σ − N +1 σ + N +1 σ − k , ρ } + 2 σ + N +1 σ − k ρσ + k σ − N +1 ] . The subindex ’sink’ emphasizes that no population canescape of site N + 1. For definitiveness and simplicity, the initial state of the network at t = 0 will be assumedto be a single excitation in site 1 unless stated otherwise.The key question that we will pose and answer is thefollowing: In a given time T , how much of the initialpopulation in site will have been transferred to the sinkat site N + 1 and how is this transfer affected by thepresence of dephasing and dissipative noise. In the remainder of this paper we will demonstratethat, in certain settings, the presence of dephasing noisecan assist the transfer of population from site 1 to thesink at site N + 1 considerably. It is an intriguing obser-vation that this noise enhanced transfer does not occurfor all possible Hamiltonians of the type given by eq.(1)and may depend also on properties of the noise such asits spatial dependence. These noise rates can be opti-mized numerically, and in very simple cases analytically,to yield the strongest possible effect. One may suspectthat natural, biological systems, have actually made useof such an optimization. Linear chain –
We begin with a brief analysis of theuniform linear chain with only nearest neighbor inter-actions so that in eq. (1) the coupling strengths sat-isfy v l,k = v k,l = vδ l,k +1 for k = 1 , . . . , N − ω k = ω and Γ k = Γ for k = 1 , . . . , N . Extensive numer-ical searches show that, for arbitrary choices of Γ N +1 ,Γ and ω and arbitrary transmission times T and chainsof the length N = 2 , . . . ,
12, the optimal choice of de-phasing noise rates vanish. We have used a directed ran-dom walk algorithm with multiple initial states whichhas never exceeded the values for the noise-free chainand approached them to within at least 10 − . We wereable to derive formulae for the case T = ∞ and shortchains which demonstrate this behaviour analytically.For N = 2, with ω = ω = ω and arbitrary v , , γ i and Γ i , we find, with the abbreviation γ = γ + γ and x = 2Γ + Γ Γ (3Γ + Γ ), that the population of thesink is given by p sink = Γ v , x + Γ (Γ + Γ ) γ + (Γ + 2Γ ) v , , (5)which is evidently maximized for γ = 0. One may alsoobtain the analytical expressions for N = 3 and Γ k = Γfor k = 1 , , γ = 0 (see section on Methods ). This ap-proach, though more tedious, may be taken to highervalues of N as well. Extensive numerical searches lendfurther support to the observation that dephasing doesnot improve excitation transfer for uniform chains but ageneral proof has remained elusive.So far, the findings are consistent with the expectationthat noise does not enhance the transport of excitations.However, for non-uniform chains we encounter the dif-ferent and perhaps surprising situation where noise cansignificantly enhance the transfer rate of excitations.As an illustrative example, we may keep the nearestneighbor coupling uniform but allow for one site to havea different site energy ω . If we chose N = 3, ω = ω = 1,Γ = Γ = Γ = 1 / v , = v , = 1 /
10, Γ N +1 = 1 / ω p s i n k ( γ op t ) − p s i n k ( γ = ) FIG. 2: The optimal improvement of the transfer efficiency isplotted versus the site frequency ω in a chain of length N = 3and system parameters ω = ω = 1, Γ = Γ = Γ = 1 / v , = v , = 1 /
10, Γ N +1 = 1 / T = ∞ . One observesthat dephasing only assists the transmission probability insome frequency intervals. and T = ∞ , then we obtain the results depicted inFig. 2. One observes that dephasing assists the trans-mission only when site 2 is sufficiently detuned from theneighboring sites. This example suggests a simple pic-ture to explain the reason for the dephasing enhancedpopulation transfer through the chain. Site 2 is stronglydetuned from its neighboring sites and the coupling v toits neighbors is comparatively weak, i.e. v ≪ δω with δω = min[ | ω − ω | , | ω − ω | ]. Hence , the transportrate is limited by a quantity of order v /δω as it is asecond order process due to the lack of resonant modesbetween neighboring sites. Introducing dephasing noiseleads to a broadening of the energy level at each site k and a line-width proportional to the dephasing rate γ k . Then, with increasing dephasing rate, the broad-ened lines of neighboring sites begin to overlap and thepopulation transfer will be enhanced as resonant modesare now available. Enhancing the dephasing rate furtherwill eventually lead to a weakening of the transfer asthe modes are distributed over a very large interval andresonant modes have a small weight. Dissipation doesnot lead to the same enhancement as, crucially, the gainto the broadening of the line is overcompensated by theirreversible loss of excitation. This is corroborated bynumerical studies where increasing dissipation does notassist the transport. The physical picture outlined aboveis confirmed in Fig. 3. We chose a chain of length 3which suffers dephasing only in site 2 and uniform dis-sipation with rates Γ k = 1 /
100 along the chain while ω = ω / ω = 1 and v , = v , = 1 /
10 (see fig.3). The close relationship of this model to Raman tran-sitions in quantum optics will be exploited to propose arealizable experiment in a highly controlled environmentto verify these effects (see section on realizations).In the examples above the improvement of excitationtransfer due to the dephasing is small. One can easily γ p s i n k ( γ ) − p s i n k ( γ = ) FIG. 3: The difference between transfer efficiency and theefficiency without dephasing is plotted versus the dephasingrate γ in a chain of length N = 3 and ω = ω / ω = 1, v , = v , = 1 / γ = γ = 0, Γ k = 1 /
100 for k = 1 , . . . , N ,Γ N +1 = 1 / T = ∞ . Initially increasing dephasing as-sists the transfer of excitation while very strong dephasingsuppresses the transport. show, however, that this improvement may be made ar-bitrarily large in the sense that without noise the trans-fer rate approaches zero while it approaches unity arbi-trarily closely for optimal noise levels. As an example,for N = 3 , ω = ω = 1; ω = 100 , v , = v , = v , γ = γ = 0 and Γ = Γ = Γ = v /f and Γ = 10 v we find for ∆ p = p sink ( γ ,opt ) − p sink ( γ = 0) thatlim v → ∆ p = f γ f γ + 3 f γ (( ω − + γ ) + (( ω − + γ ) (6)This is maximized for γ = ω − p = f / ( f + 6 f ( ω −
1) + 4( ω − − ). In thelimit f → ∞ this approaches 1, that is, without noise theexcitation transfer vanishes while with noise it achievesunit efficiency! It should be noted that being a system offixed finite size, the effect may not be directly attributedto Anderson localization [24] which, in addition does notoccur in systems attached to a sink, as is assumed here[25]. Entanglement and coherence in the channel –
We haveseen that the transport of excitations in the system maybe assisted considerably by local dephasing. Now wewould like to discuss briefly the quantum coherence prop-erties during transmission by studying the presence ofentanglement and the ability of the chain to transmitquantum information. To this end, we consider how en-tanglement is transported along the chain when it is usedto propagate one half of a maximally entangled state toobtain an insight on how is the quantum capacity of thischannel affected by dephasing. To illustrate this, we con-sider a chain of N = 4 sites. We chose the same param-eters as in Fig. 2 and fix ω = 14. Comparison of theentanglement between an uncoupled site and the varioussites in the chain for vanishing dephasing and the opti-mal choice of the dephasing for excitation transfer showthat, while entanglement propagates through the system,the amount of entanglement decreases with increasingdephasing. In fact, the dephasing rate that optimizesthe ability of the channel to transmit quantum informa-tion vanishes, in contrast to the situation for excitationtransfer. Therefore, although dephasing may enhancethe propagation of excitations, it also destroys quantumcoherence and in the present setting it leaves an overalldetrimental effect. Complex networks and Light-harvesting molecules –
So far, we have demonstrated that in linear chains lo-cal dephasing noise may enhance the transfer of excita-tions. Going beyond this, we will now consider fully con-nected networks and apply our observations to a modelthat describes the transfer of excitons in the Fenna-Matthews-Olson complex of
Prosthecochloris aestuarii ,which is a pigment-protein complex that consists of sevenbacteriochlorophyll-a (BChla) molecules (see [20, 21, 22]and Note Added for closely related work). This complexis able to absorb light to create an exciton. This exci-ton then propagates through the complex until it reachesthe reaction centre where its energy is then used to trig-ger further processes that bind the energy in chemicalform [15, 23]. The Hamiltonian of this complex may beapproximated by eq. (1), where the site energies andcoupling constants may be taken from table 2 and 4 of[15]. We then find, in matrix form H = − . . − . . − . − . − . . . . . . . . . . − . . − . . − . . − . . − . − . − . . . . − . . . − . − . . − . − . . . . − . . . − . − . . . (7)where we have shifted the zero of energy by12230 (all number are given in the units of1 . · − N m = 1 . − eV ) for all sitescorresponding to a wavelength of ∼ = 800 nm . Recentwork [15] suggests that it is this site 3 that couples tothe reaction centre at site 8. For this rate, somewhatarbitrarily, we chose Γ , = 10 / .
88 correspondingto about 1 ps − (value in the literature range from0.25 ps − [15] and 1 ps − [20] to 4 ps − [17]). Again, wewill assume the presence of both dissipative noise (lossof excitons) and dephasing noise (due to the presenceof a phonon bath consisting of vibrational modes of themolecule). The measured lifetime of excitons is of theorder of 1 ns which determines a dissipative decay rateof 2Γ k = 1 /
188 and that we assume to be the same foreach site [15]. If we neglect the presence of any formof dephasing and we start with a single excitation onsite 1, then we observe that the excitation is transferredto the reaction centre (site 8). For a time T = 5, wefind that the amount of excitation that is transferredis p sink = 0 . T = 5 we find the t E N FIG. 4: The time evolution of the entanglement between adecoupled site and the sites in the chain of length N = 4and system parameters ω = ω = ω = 10, ω = 14, v , = v , = v , = 1, Γ k = 1 /
10 for k = 1 , . . . , N and Γ N +1 =1. The initial state is a maximally entangled state betweenthe decoupled site and the first site of the chain. Dephasingdestroys entanglement along the chain and has no beneficialeffect. optimal dephasing rates ( γ , γ , γ , γ , γ , γ , γ ) =(469 . , . , . , . , . , . , .
08) andthe much improved value p sink = 0 . T = ∞ , we find the dephasing free transferprobability of p sink = 0 . γ , γ , γ , γ , γ , γ , γ ) =(27 . , . , . , . , . , . , .
35) we find p sink = 0 . T = 5 [15]. It is remarkable thatsuch a rapid transfer cannot be explained from a purelycoherent dynamics and, as shown above, the underlyingreason for the speed up is the presence of dephasing whichmay even be local. Experimental Realizations –
The FMO-complex pro-vides a fascinating setting for the observation of dephas-ing enhanced transport but it is also a very challeng-ing environment to verify the effect precisely. Here wepresent several physical systems in which the dephasingenhanced excitation transfer may be observed and whichare at the same time highly controllable. Perhaps thesimplest such setting is found in atomic physics (see Fig.5) where the behaviour of a chain of three sites may besimulated using detuned Raman transitions in ions suchas Ca + , Sr + or Ba + . The master equation of this systemsimulates exactly that of a chain with a single excitationas has been described throughout this paper. Atomic WW r S G N + G WW r S G N + G WW r S G N + G FIG. 5: A atomic system with Raman transitions provides atransparent illustration of dephasing assisted transport. Therequired level structure may be realized in Ca + , Sr + or Ba + .Each atomic level represents a site in the chain which maybe populated. Starting with all the population in level 1,one may then irradiate the system with classical laser fieldsof Rabi-frequency Ω on the 1 ↔ ↔ | r i that plays the role of the recipient.Spontaneous decay of the chain as a whole is modelled byspontaneous decay into level | i from which no populationcan enter the levels | i , | i , | i and | r i anymore. Dephasingnoise may now enter the system affecting level 2 for examplethrough magnetic field fluctuations. populations may be measured with very high accuracyusing quantum jump detection [28, 29].A variety of other natural implementations of dephas-ing assisted excitation transport can be conceived andwill be studied in detail elsewhere. Firstly, the oscilla-tions of ions in a linear ion trap transversal to the trapaxis realizes a harmonic chain [30] that allows for theimplementation of a variety of operations such as prepa-ration of Fock states and is capable of supporting near-est neighbor coupling between neighbouring ion oscilla-tors [31] and allowing high efficiency readout by quantumjump detection [28]. When restricting to the single exci-tation space, the dynamics of the system is described bymaster equations that become equivalent to those pre-sented in this paper.Furthermore, harmonic chains are also realized in cou-pled arrays of cavities which have recently received con-siderable attention in the context of quantum simulators[32]. Ultra-cold atoms in optical lattices which have pre-viously been used to study thermal assisted transport inBrownian ratchets [33] presents another scenario in whichto study such dephasing assisted processes. Chains of su-perconducting qubits or superconducting stripline cavi-ties [34] may also provide a possible setting for the ob-servation of the effects described above. Conclusions and outlook –
The results presented heredemonstrate that while dephasing noise destroys quan-tum correlations, it may at the same time enhance thetransport of excitations. In fact, the efficient transportobserved in certain biological systems has been shownto be incompatible with a fully coherent evolution while it can be explained if the system is subject to local de-phasing. Hence, in this context, the presence of quan-tum coherence and therefore, entanglement in the sys-tem, does not seem to be supporting excitation transfer.This suggests that entanglement that may be present inbio-molecules, though interesting, may not be a universalfunctional resource.Importantly, the results presented here suggest that itmay be possible to design and optimize the performanceof nano-fabricated transmission lines in naturally noisyenvironments to achieve strongly enhanced transfer ef-ficiencies employing the concept of noise assisted trans-port.
Acknowledgements–
We are grateful to Seth Lloydfor helpful communications concerning [20, 21, 22], NeilOxtoby, Angel Rivas and Shashank Virmani for usefulcomments on the manuscript and to Danny Segal foradvice on atomic physics. This work was supportedby the EU via the Integrated Project QAP (‘QubitApplications’) and the STREP action CORNER and theEPSRC through the QIP-IRC. MBP holds a WolfsonResearch Merit Award.
Note Added—
While finalizing this work, we becameaware of independently obtained but closely related re-sults presented in [20, 21, 22]. There it was showed thatquantum transport can be enhanced by an interplay be-tween coherent dynamics and environment effects withparticular emphasis on excitonic energy transfer in lightharvesting complexes [20]. The role of the different phys-ical processes that contribute to the energy transfer ef-ficiency have been studied in [21] and the enhancementof quantum transport due to a pure dephasing environ-ment within the Haaken-Strobl model was demonstratedin [22].
Methods –Exact solutions for uniform chains –
One may also ob-tain the analytical expressions for a chain of length N = 3 described by eqs. (1) - (4) for the choice and Γ k = Γ for k = 1 , , , γ = 0. We find p sink = (4Γ + γ + γ ) v + 6 a Γ + 2Γ (3 γ + 3 γ + 8 b + 2 γ + 32 v ) + Γ (2 c + dv ) + Γ v (3 γ + 7 b + 4 γ + 15 v ) + 4( γ + γ ) v where a = (5 γ + 5 γ + 4 γ ), b = γ γ + γ γ + γ γ , c = γ ( γ + γ ) + γ ( γ + γ ) + γ ( γ + γ ) + 2 γ γ γ , d = 32 γ + 25 γ + 29 γ . Then one first observes thatthe optimal choice is γ = 0 as it only occurs in thedenominator with positive coefficients. In the remainingexpression one then substitutes γ k = ˜ γ k allowing also fornegative ˜ γ k . Then differentiation w.r.t these ˜ γ k showsthat the gradient only vanishes for ˜ γ = ˜ γ = 0. Beyond Markovian master equations–
So far we havedemonstrated the existence of dephasing enhanced exci-tation transfer employing a master equation description.The optimized dephasing rates that have been obtained,in particular those in the context of the FMO complex,can be comparable to the coherent interaction strengthsand may be similar to the spectral width of the bath re-sponsible for the dephasing [15]. This may not be fullycompatible with the master equation approach employedso far as its derivation relies on several assumptions in-cluding the weak coupling hypothesis and the require-ment for the bath to be Markovian [26]. The derivation isfurther complicated for systems with several constituentswhere the local coupling of its constituents is not com-patible with non-local structure of the eigenmodes of thesystems. This is especially so when the coherent intersub-system coupling is of comparable strength to the sys-tem environment coupling. The situation is made moredifficult due to spatial as well as temporal correlations inthe environmental noise (which is to be expected in par-ticular for the FMO complex but also many other realisa-tions of coupled chains in contact with an environment).Bloch-Redfield equations and other effective descriptionare sometimes used but still represent approximations tothe correct dynamics [26] where the errors are often dif-ficult to estimate precisely.Therefore, we demonstrate briefly that dephasing as-sisted transfer of excitation can also be observed whenone uses a microscopic model of an environment thatmay, in addition, exhibit non-Markovian behaviour. Tothis end we study the effect of an environment which ismodelled by brief interactions between two-level systemsand individual subsystem of the chain in which excita- tion transport is taking place. The strength and natureof the interactions can be chosen to implement dephasing(elastic collisions) and dissipation (in-elastic collisions).Non-markovian effects can be included in the model de-pending on the spatial and temporal memory of the envi-ronment particles. Interaction strengths are determinedfor a single site system to obtain the dissipation rate Γand dephasing rate γ . This simplified model allows usto study the effect of more realistic environments outsidethe master equation picture and results are summarizedin Figure 6. A more detailed simulation of excitationtransfer taking account of the full environment are be-yond the scope of the present work and will be presentedelsewhere [35] t p s i n k γ =0 γ =0.0064 γ opt γ = 0.16 γ opt γ = γ opt FIG. 6: Here we show how the transfer in the presence ofdephasing into a bath that is modelled by a collisional modelwhere local sites briefly interact with a single particle. The in-teraction strength is chosen such that in an uncoupled systemsthe sites suffer the optimal decoherence rates γ opt as presentedin the previous section multiplied with factors 0 , . , . , 292 (2008). [2] R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A , L453 (1981).[3] L. Gammaitoni, P. H¨anggi, P. Jung, and F. Marchesoni,Rev. Mod. Phys. , 223 (1998).[4] J. K Douglass, L. Wilkens, E. Pantazelou and F. Moss,Nature , 337 (1993); K. Wiesenfeld and F. Moss, Na-ture , 33 (1995).[5] Y. H. Shang, A. Claridge-Chang, L. Sjulson, M. Pypaertand G. Miesenbock, Cell , 601 (2007).[6] P. Reimann, M. Grifoni, and P. H¨anggi, Phys. Rev. Lett. , 10 (1997)[7] Special issue on Ratchets and Brownian Motors, editedby H. Linke, Appl. Phys. A , 167 (2002)[8] P. H¨anggi, F. Marchesoni and F. Nori, Ann. Phys.(Berlin) , 51 (2005).[9] L. Viola, E. M. Fortunato, S. Lloyd, C. H. Tseng, andD. G. Cory, Phys. Rev. Lett. , 5466 (2000).[10] M. B. Plenio and S. F. Huelga, Phys. Rev. Lett. ,197901 (2002).[11] S. F. Huelga and M. B. Plenio, Phys. Rev. Lett. ,170601 (2007).[12] J.-L. Ting, Phys. Rev. E , 2801 (1998)[13] G. Bowen and S. Mancini, Phys. Lett. A , 1 (2004)´ıbid , 272 (2006).[14] C. Di Franco, M. Paternostro, D. I. Tsomokos and S. F.Huelga, Phys. Rev. A , 062337 (2008).[15] J. Adolphs and T. Renger, Biophys. J. , 2778 (2006)[16] See G. S. Engel, T. R. Calhoun, E. L. Read EL, T. K.Ahn, T. Mancal, Y. C. Cheng, R. E. Blankenship andG. R. Fleming, Nature , 782 (2007) for recent exper-imental results.[17] A. Olaya-Castro, C. F. Lee, F. Fassioli-Olsen, and N. F.Johnson, arXiv:0708.1159.[18] H. J. Briegel and S. Popescu, arXiv:0806.4552.[19] K. M. Gaab and C. J. Bardeen, J. Chem. Phys. ,7813 (2004).[20] M. Mohseni, P. Rebentrost, S. Lloyd and A. Aspuru- Guzik, arXiv:0805.2741.[21] P. Rebentrost, M. Mohseni and A. Aspuru-Guzik,arXiv:0806.4725.[22] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, A.Aspuru-Guzik, arXiv:0807.0929[23] R. E. Fenna and B. W. Matthews, Nature , 573(1975).[24] P. W. Anderson, Phys. Rev. , 1492 (1958).[25] S. A. Gurvitz, Phys. Rev. Lett. , 812 (2000).[26] H.-P. Breuer and F. Petruccione, The Theory of OpenQuantum Systems , Oxford University Press, 2002.[27] M. O. Scully and S. Zubairy,
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