Photosynthesis is a biological process that involves the highly-efficient
transport of energy captured from the sun to a reaction center, where
conversion into useful biochemical energy takes place. Even though one can
always use a quantum perspective to describe any physical process, since
everything follows the laws of Quantum Mechanics, is the use of quantum theory
imperative to explain this high efficiency? Making use of the quantum-classical
correspondence of electronic energy transfer recently introduced by Eisfeld and
Briggs [Phys. Rev. E 85, 046118 (2012)], we show here that the highly-efficient
noise-assisted energy transport described by Rebentrost et al. [New J. Phys.
11, 033003 (2009)], and Plenio and Huelga [New J. Phys. 10, 113019 (2008)], as
the result of the interplay between the quantum coherent evolution of the
photosynthetic system and noise introduced by its surrounding environment, it
can be found as well in purely classical systems. The wider scope of
applicability of the enhancement of energy transfer assisted by noise might
open new ways for developing new technologies aimed at enhancing the efficiency
of a myriad of energy transfer systems, from information channels in
micro-electronic circuits to long-distance high-voltage electrical lines.Comment: 4 pages, 3 figure

León-Montiel, R. de J.

Torres, Juan P.

English

arXiv

R. de J. León-Montiel

Juan P. Torres

Crossref

Physical Review Letters

Highly Efficient Noise-Assisted Energy Transport in Classical Oscillator Systems

Photosynthesis is a biological process that involves the highly-efficient transport of energy captured from the sun to a reaction center, where conversion into useful biochemical energy takes place. Even though one can always use a quantum perspective to describe any physical process, since everything follows the laws of Quantum Mechanics, is the use of quantum theory imperative to explain this high efficiency? Making use of the quantum-classical correspondence of electronic energy transfer recently introduced by Eisfeld and Briggs [Phys. Rev. E 85, 046118 (2012)], we show here that the highly-efficient noise-assisted energy transport described by Rebentrost et al. [New J. Phys. 11, 033003 (2009)], and Plenio and Huelga [New J. Phys. 10, 113019 (2008)], as the result of the interplay between the quantum coherent evolution of the photosynthetic system and noise introduced by its surrounding environment, it can be found as well in purely classical systems. The wider scope of applicability of the enhancement of energy transfer assisted by noise might open new ways for developing new technologies aimed at enhancing the efficiency of a myriad of energy transfer systems, from information channels in micro-electronic circuits to long-distance high-voltage electrical lines.Peer ReviewedPostprint (published version

Leon Montiel, R de J.

Pérez Torres, Juan

UPCommons. Portal del coneixement obert de la UPC

Highly Efficient Noise-Assisted Energy Transport in Classical Oscillator SystemsR. de J. Leo´n-Montiel1,* and Juan P. Torres1,2,†1ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain2Department of Signal Theory and Communications, Campus Nord D3, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain(Received 7 January 2013; published 21 May 2013)Photosynthesis is a biological process that involves the highly efficient transport of energy capturedfrom the Sun to a reaction center, where conversion into useful biochemical energy takes place. Using aquantum description, Rebentrost et al. [New J. Phys. 11, 033003 (2009)] and Plenio and Huelga [New J.Phys. 10, 113019 (2008)] have explained this high efficiency as the result of the interplay between thequantum coherent evolution of the photosynthetic system and noise introduced by its surroundingenvironment. Even though one can always use a quantum perspective to describe any physical process,since everything follows the laws of quantum mechanics, is the use of quantum theory imperative toexplain this high efficiency? Recently, it has been shown by Eisfeld and Briggs [Phys. Rev. E 85, 046118(2012)] that a purely classical model can be used to explain main aspects of the energy transfer inphotosynthetic systems. Using this approach, we demonstrate explicitly here that highly efficient noise-assisted energy transport can be found as well in purely classical systems. The wider scope of applicabilityof the enhancement of energy transfer assisted by noise might open new ways for developing newtechnologies aimed at enhancing the efficiency of a myriad of energy transfer systems, from informationchannels in microelectronic circuits to long-distance high-voltage electrical lines.DOI: 10.1103/PhysRevLett.110.218101 PACS numbers: 87.10.e, 05.40.Ca, 82.20.Nk, 82.20.RpBecause of its undoubted importance for all life onEarth, molecular mechanisms of energy transport in photo-synthetic light-harvesting complexes have been a subjectof study for decades [1–4]. In recent years, a renewedinterest on this topic has arisen [5,6], mainly due to theunexpected observation of long-lived electronic coher-ences in the energy transfer process of photosyntheticsystems, particularly in the nowadays most widely inves-tigated system, the Fenna-Matthews-Olson (FMO)complex [7–9].As a consequence of these findings, several theoreticalstudies have been devoted to describing how coherenceeffects in a quantum scenario might play an important rolein the remarkably high efficiency of energy transfer inphotosynthetic systems [10–12]. This is especially notablesince it takes place in a scenario apparently not propitiousfor the observation of quantum effects. In particular, it hasbeen suggested that high efficiency transport arises as aresult of the dynamical interplay between the quantumcoherent evolution of the photosynthetic system and thedephasing noise introduced by its surrounding environ-ment, a phenomenon called environment-assisted quantumtransport (ENAQT) [13] or dephasing-assisted energytransport [14].As stated in Ref. [15], ENAQT can be understood as thesuppression of coherent quantum localization mediated bynoise, helping the excitation to move faster through thephotosynthetic system, thus increasing the efficiency ofenergy transport. In this way, ENAQT might be seen as aphenomenon that exists only in a regime where the quan-tum and classical worlds overlap. Notwithstanding, makinguse of the quantum-classical correspondence of electronicenergy transfer presented in Ref. [16], we show here thatthe same effect can also be found in purely classicalsystems. Our departure point is based on the considerationthat aggregates of coupled monomers (such as the FMOcomplex) can also be described as a system of weaklyinteracting classical oscillators [17]. We then demonstratethat the noise-assisted enhancement of transport efficiencyin the FMO complex, shown in Refs. [13,14], and based ona pure quantum formalism, can also be found in a purelyclassical model, without the need to resort to quantumeffects.For the sake of comparison and clarity, we will firstmodel the FMO complex as a quantum system of Ninteracting sites, where the interaction of each site withits surrounding environment is modeled by a pure dephas-ing process. We have adopted this model because of itsextended use for describing noise-assisted energy transferprocesses in photosynthetic systems [13,14]. Next, wewill present the classical model of Refs. [16,17], whichcorresponds to a system of N weakly coupled harmonicclassical oscillators. In this case, environmental effects areintroduced by assuming that the frequency of each oscil-lator varies stochastically as a Gaussian Markov process.Finally, we will solve both models using the siteenergies and coupling coefficients for the FMO complexof Prosthecochloris aestuarii to show that the sameenvironment-assisted energy transfer effect can be foundin both classical and quantum models.The Hamiltonian of a system comprising N interactingsites in the presence of a single excitation is given byPRL 110, 218101 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending24 MAY 20130031-9007=13=110(21)=218101(4) 218101-1  2013 American Physical SocietyH^ S ¼XNn¼1njnihnj þXNnmVnmjnihmj; (1)where jni denotes the excitation being at site n. Thenth-site energies and the coupling between sites n and mare described by n and Vnm, respectively.We make use of a simple model where the dynamics ofthe system interacting with a surrounding environment isdescribed by a Lindblad master equation, which in theBorn-Markov and secular approximations is written as [18]@^nm@t¼  i@½H^S; ^nm þ L^deph½^nm þ D^½^nm: (2)Here, the interaction of the system with the environmentis characterized by a pure dephasing process given bythe Lindblad operator L^deph½^nm ¼ ½1=2ðn þ mÞ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃnmpnm^nm, with n being the dephasing rates.Although the pure dephasing model is not able to captureimportant aspects of electronic energy transfer, such asphonon relaxation [19], it provides a useful description ofenvironmental effects in a simple way. To quantify thetransfer of energy from a chosen site k to the reactioncenter, we have phenomenologically introduced an irre-versible decay process (with rate ) described by theoperator D^, which is given by [20] D^½^nm ¼ fjkihkj; ^gnm, where f  g stands for the anticommutator.Making use of Eq. (2), one can define a measure for theefficiency of energy transport as the population transferredto the reaction center, within a time t, asQeff ¼ 2Z t0hkj^ðsÞjkids: (3)Equations (2) and (3) constitute the quantum equations,which have to be compared with the equations that will beobtained in the classical model.For the classical case, we consider an ensemble of Ncoupled harmonic oscillators, each with mass M and fre-quency !n. The temporal evolution of the system isdescribed by a classical Hamiltonian, which in termsof the position qn and momentum pn of each oscillatorreads asHS ¼Xnp2n2MþM!2n2q2nþ 12XnmKnmqnqm; (4)where Knm stands for the coupling coefficient betweenthe oscillators. By defining a new dimensionless com-plex amplitude [21], ~znðtÞ ¼ ~qnðtÞ þ i~pnðtÞ, with ~qn ¼ðM!n=2@Þ1=2qn and ~pn ¼ ð2@M!nÞ1=2pn, the Hamiltonequations of motion of the system can be cast into a singleequation:@~zn@t¼ i!n~zn  iXm~Knm Ref~zmg: (5)Ref  g stands for the real part of a complex number and~Knm ¼ Knm=ðM ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!n!mp Þ.To include environmental effects, we proceed in thesame manner as in the construction of a Kubo oscillator[22,23]. For this, we assume that the frequency of eachclassical oscillator varies randomly as a stochastic process:!nðtÞ ¼ !n þnðtÞ. !n is now the average frequencyof the nth oscillator and nðtÞ describes a GaussianMarkov process with zero average (Wiener process), i.e.,hnðtÞi ¼ 0 and hnðtÞmðt0Þi ¼ nnmðt t0Þ, whereh  i denotes stochastic averaging.In Ref. [16], it has been shown that one can transformEq. (5), within the framework of Itoˆ calculus [24], into aclassical master equation that describes the temporal dy-namics of the system of coupled harmonic oscillators,when interaction with the surrounding environment istaken into account. To describe the transfer of excitationfrom the kth oscillator to the reaction center, we extend thisresult, and introduce an irreversible decay process (withrate ), described by D½nm ¼ fjkihkj; gnm, wherenm ¼ h~zn~zmi. In this way, we can write the classicalmaster equation as@nm@t¼H ½nm þL½nm þD½nmþ i@XjðVmjh~zj~zni  Vnjh~zj~zmiÞ; (6)with Vnm ¼ ~Knm@=2, andH ½nm ¼ ið!n !mÞnm i@XjðVnjjm  VjmnjÞ; (7)L ½nm ¼ 12ðn þ mÞ  ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃnmp nmnm: (8)The energy transfer efficiency within the ensemble ofoscillators is then given byCeff ¼ 2Z t0kkðsÞds: (9)ðtÞ ¼ ðtÞ=Pnnn is the normalized classical densityoperator.Equations (6) and (9) represent the classical equations,whose results have to be compared with their quantum-mechanical counterpart, Eqs. (2) and (3). To this end, wecan make use of the site energies and coupling coefficientsfor the FMO complex of P. aestuarii [25]. The FMO is apigment-protein complex that guides the energy from thelight-harvesting chlorosomes to the reaction center ingreen sulfur bacteria [26,27]. It is a trimer of three identicalsubunits interacting weakly with each other. Each subunitis composed of seven bacteriochlorophyll-a (BChla) mole-cules embedded in a scaffolding of protein molecules, asshown in Fig. 1. The FMO complex is generally modeledPRL 110, 218101 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending24 MAY 2013218101-2by a network of seven different sites, where the dynamicsof a single excitation through the complex is governed bythe specific values of the site energies (n) and the couplingcoefficients (Vnm). In particular, we will use the values ofthe site energies and coupling coefficients for P. aestuarii,as stated in Tables 2 and 4 of Ref. [25]. The initial state ofthe system corresponds to a single excitation in site 1. Inthe FMO, the BChl 3 is in the vicinity of the reaction center[25]. Thus, we take this site (k ¼ 3) as the main excitationdonor to the reaction center, with a transfer rate estimatedto be  ¼ 1 ps1 [28]. Furthermore, for the pure dephasingprocess, we consider that dephasing rates are the same forall sites ( ¼ n) and that the efficiency of energy transferis limited by the finite excitation lifetime (t 1 ns).Figure 2 shows the efficiency of energy transfer as afunction of the dephasing rate  obtained by means of thequantum equations (2) and (3). Notice that at low dephas-ing, i.e., with environment effects not considered, coherentevolution of the system leads to an efficiency of about 90%.When increasing the dephasing, efficiency grows to almost100%, showing that the environment affects the system insuch a way that it becomes more efficient for transferringenergy to the reaction center. Finally, for stronger dephas-ing, efficiency drops rapidly and almost no energy is trans-ferred to the reaction center. Qualitatively similar resultshave also been obtained for the case of the FMO complexof Chlorobium tepidum [13].We now turn our attention to the case of the classicalmodel by solving Eqs. (6) and (9). Figure 3 shows theefficiency of the energy transfer as a function of the dephas-ing rate. We observe that the same noise-assisted effect isalso present in the purely classical model. For the sake ofcomparison, Fig. 3 also shows the solution of the quantum-mechanical model (dashed line). Notice that both solutionsagree for dephasing rates up to 103 ps1. However, forlarger values of dephasing the quantum and classical solu-tions differ from each other. This is in agreement with thefact that both solutions are the same, provided that thecondition  !n is satisfied [16].Noise-assisted energy transport in disordered systemshas been understood as the suppression of coherent quan-tum localization through noise, bringing the detuned quan-tum levels into resonance and thus facilitating the energytransfer [13,15]. Notwithstanding, the results presentedhere show that the same effect can also be found in purelyclassical systems. This implies that one can make use ofsuch systems in order to simulate the intricate energytransfer mechanisms that take place in molecular aggre-gates, such as the photosynthetic FMO complex.Recently, it has been suggested that classical LC circuitoscillators (where L stands for inductance and C forcapacitance) can be used to model coupled quantum two-level systems [29]. Hence, one could devise an experimen-tal apparatus comprising eight electrical oscillators withthe eighth acting as the reaction center, which would beFIG. 2 (color online). Energy transfer efficiency as a functionof the dephasing rate  obtained from the quantum-mechanicalequations.FIG. 3 (color online). Energy transfer efficiency as a functionof the dephasing rate  obtained from the classical equations(solid line). For the sake of comparison, we have also includedhere the curve shown in Fig. 2, which corresponds to the solutionof the quantum equations (dashed line).FIG. 1 (color online). Arrangement of the BChla molecules ofa single unit of the Fenna-Matthews-Olson (FMO) complex. Thefigure was created using PyMOL [35], and is based on theProtein Data Bank entry 3ENI.PRL 110, 218101 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending24 MAY 2013218101-3strongly coupled to one of the remaining oscillators. Then,by stochastically modulating the frequencies!n, and prop-erly controlling the noise intensity n, one would be able toobserve the noise-assisted energy transfer phenomenon bymonitoring the signal present in the eighth oscillator. Theseclassical simulations could be further used to compare withthe recent experimental proposal of noise-assisted trans-port based on coupled quantum-optical cavities [30].The concept of noise-assisted energy transport has beenextensively used for describing the inner working of quan-tum and classical systems [31]. Along these lines, theparticular enhancement effect described in this Lettermight open a new research direction towards new methodsfor enhancing the efficiency of a myriad of energy trans-port systems that inevitably live in a noisy environment,from small-scale information and energy transfer systemsin microwave and photonic circuits to long-distance high-voltage electrical lines. In this way, a specific featureinitially conceived in a quantum scenario (environment-assisted energy transport) is shown to arise as well in apurely classical context, widening thus the scope of pos-sible quantum-inspired technological applications.To conclude, the search and demonstration of systems inwhich to observe quantum-mechanical effects with noclassical counterpart is a subject of lively interest anddebate [32–34]. Biological systems are not, in principle,a propitious scenario for the observation of quantum fea-tures, such as quantum superposition, interference, orentanglement. Nevertheless, one can always use a quantumperspective to describe any physical process, since every-thing follows the laws of quantum mechanics. This doesnot mean, however, that in certain cases a purely classicalmodel may not similarly reproduce some of the resultspredicted by the full quantum-mechanical model, sinceclassical physics emerge, after all, from quantum physicsunder many circumstances.This work was supported by projects FIS2010-14831and FET-Open Grant No. 255914 (PHORBITECH). Thiswork has also been partially supported by FundacioPrivada Cellex Barcelona.*roberto.leon@icfo.es†juanp.torres@icfo.es[1] J. Frenkel, Phys. Rev. 37, 17 (1931).[2] J. Franck and E. Teller, J. Chem. Phys. 6, 861 (1938).[3] T. Fo¨rster, Modern Quantum Chemistry (Academic,New York, 1965).[4] R. E. Blankenship, Molecular Mechanisms ofPhotosynthesis (Blackwell, Oxford, 2002).[5] P. Ball, Nature (London) 474, 272 (2011).[6] N. Lambert, Y.N. Chen, Y. C. Cheng, C.M. Li, G. Y.Chen, and F. Nori, Nat. Phys. 9, 10 (2013).[7] G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T.Mancˇal, Y.-C. Chen, R. E. Blankenship, and G. R.Fleming, Nature (London) 446, 782 (2007).[8] G. Panitchayangkoon, D. Hayes, K. A. Fransted, J. R.Caram, E. Harel, J. Wen, R. E. Blankenship, and G. S.Engel, Proc. Natl. Acad. Sci. U.S.A. 107, 12 766 (2010).[9] E. Collini, C. Y. Wong, K. E. Wilk, P.M.G. Curmi, P.Brumer, and G.D. Scholes, Nature (London) 463, 644(2010).[10] A. Ishizaki and G. R. Fleming, J. Chem. Phys. 130,234111 (2009).[11] S. Hoyer, M. Sarovar, and K. B. Whaley, New J. Phys. 12,065041 (2010).[12] F. Fassioli and A. Olaya-Castro, New J. Phys. 12, 085006(2010).[13] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, and A.Aspuru-Guzik, New J. Phys. 11, 033003 (2009).[14] M. Plenio and S. Huelga, New J. Phys. 10, 113019(2008).[15] I. Kassal and A. Aspuru-Guzik, New J. Phys. 14, 053041(2012).[16] A. Eisfeld and J. S. Briggs, Phys. Rev. E 85, 046118 (2012).[17] J. S. Briggs and A. Eisfeld, Phys. Rev. E 83, 051911(2011).[18] H.-P. Breuer and F. Petruccione, The Theory of OpenQuantum Systems (Oxford University, New York, 2002).[19] A. Ishizaki and G. R. Fleming, J. Chem. Phys. 130,234110 (2009).[20] M. Scully and M. S. Zubairy, Quantum Optics (CambridgeUniversity Press, Cambridge, England, 2006).[21] F. Strocchi, Rev. Mod. Phys. 38, 36 (1966).[22] R. Kubo, J. Math. Phys. (N.Y.) 4, 174 (1963).[23] R. F. Fox, Phys. Rep. 48, 179 (1978).[24] N. G. van Kampen, J. Stat. Phys. 24, 175 (1981).[25] J. Adolphs and T. Renger, Biophys. J. 91, 2778 (2006).[26] R. E. Fenna and B.W. Matthews, Nature (London) 258,573 (1975).[27] C. Sybesma and J.M. Olson, Proc. Natl. Acad. Sci. U.S.A.49, 248 (1963).[28] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, J. Chem. Phys. 129, 174106 (2008).[29] J. S. Briggs and A. Eisfeld, Phys. Rev. A 85, 052111(2012).[30] F. Caruso, N. Spagnolo, C. Vitelli, F. Sciarrino, and M. B.Plenio, Phys. Rev. A 83, 013811 (2011).[31] P. Ha¨nggi and F. Marchesoni, Rev. Mod. Phys. 81, 387(2009).[32] E. N. Zimanyi and R. J. Silbey, J. Chem. Phys. 133,144107 (2010).[33] W.H. Miller, J. Chem. Phys. 136, 210901 (2012).[34] M. Tiersch, S. Popescu, and H. J. Briegel, Phil. Trans. R.Soc. A 370, 3771 (2012).[35] Schro¨dinger LLC, The PyMOL Molecular GraphicsSystem (version 1.4.1).PRL 110, 218101 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending24 MAY 2013218101-4

Highly efficient noise-assisted energy transport in classical oscillator systems

Photosynthesis is a biological process that involves the highly-efficient transport of energy captured from the sun to a reaction center, where conversion into useful biochemical energy takes place. Even though one can always use a quantum perspective to describe any physical process, since everything follows the laws of Quantum Mechanics, is the use of quantum theory imperative to explain this high efficiency? Making use of the quantum-classical correspondence of electronic energy transfer recently introduced by Eisfeld and Briggs [Phys. Rev. E 85, 046118 (2012)], we show here that the highly-efficient noise-assisted energy transport described by Rebentrost et al. [New J. Phys. 11, 033003 (2009)], and Plenio and Huelga [New J. Phys. 10, 113019 (2008)], as the result of the interplay between the quantum coherent evolution of the photosynthetic system and noise introduced by its surrounding environment, it can be found as well in purely classical systems. The wider scope of applicability of the enhancement of energy transfer assisted by noise might open new ways for developing new technologies aimed at enhancing the efficiency of a myriad of energy transfer systems, from information channels in micro-electronic circuits to long-distance high-voltage electrical lines.Peer Reviewe

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https://upcommons.upc.edu/bitstream/2117/19952/1/PhysRevLett.110.218101.pdf

Highly-efficient noise-assisted energy transport in classical oscillator systems

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