Highly efficient noise-assisted energy transport in classical oscillator systems

Highly E icien Noise-Assis ed Ene gy T anspo in Classical Oscilla o Sys ems R. de J. Leo ´n-Mon iel 1, *and Juan P. To es 1,2,† 1 ICFO-Ins i u de Ciencies Fo oniques, Medi e anean Technology Pa k, 08860 Cas ellde els (Ba celona), Spain 2 Depa men o Signal Theo y and Communica ions, Campus No d D3, Uni e si a Poli ecnica de Ca alunya, 08034 Ba celona, Spain (Recei ed 7 Janua y 2013; published 21 May 2013) Pho osyn hesis is a biological p ocess ha in ol es he highly e icien anspo o ene gy cap u ed om he Sun o a eac ion cen e , whe e con e sion in o use ul biochemical ene gy akes place. Using a quan um desc ip ion, Reben os e al. [New J. Phys. 11, 033003 (2009)] and Plenio and Huelga [New J. Phys. 10, 113019 (2008)] ha e explained his high e iciency as he esul o he in e play be ween he quan um cohe en e olu ion o he pho osyn he ic sys em and noise in oduced by i s su ounding en i onmen . E en hough one can always use a quan um pe spec i e o desc ibe any physical p ocess, since e e y hing ollows he laws o quan um mechanics, is he use o quan um heo y impe a i e o explain his high e iciency? Recen ly, i has been shown by Eis eld and B iggs [Phys. Re . E 85, 046118 (2012)] ha a pu ely classical model can be used o explain main aspec s o he ene gy ans e in pho osyn he ic sys ems. Using his app oach, we demons a e explici ly he e ha highly e icien noise- assis ed ene gy anspo can be ound as well in pu ely classical sys ems. The wide scope o applicabili y o he enhancemen o ene gy ans e assis ed by noise migh open new ways o de eloping new echnologies aimed a enhancing he e iciency o a my iad o ene gy ans e sys ems, om in o ma ion channels in mic oelec onic ci cui s o long-dis ance high- ol age elec ical lines. DOI: 10.1103/PhysRe Le .110.218101 PACS numbe s: 87.10.e, 05.40.Ca, 82.20.Nk, 82.20.Rp Because o i s undoub ed impo ance o all li e on Ea h, molecula mechanisms o ene gy anspo in pho o- syn he ic ligh -ha es ing complexes ha e been a subjec o s udy o decades [1–4]. In ecen yea s, a enewed in e es on his opic has a isen [5,6], mainly due o he unexpec ed obse a ion o long-li ed elec onic cohe - ences in he ene gy ans e p ocess o pho osyn he ic sys ems, pa icula ly in he nowadays mos widely in es- iga ed sys em, he Fenna-Ma hews-Olson (FMO) complex [7–9]. As a consequence o hese indings, se e al heo e ical s udies ha e been de o ed o desc ibing how cohe ence e ec s in a quan um scena io migh play an impo an ole in he ema kably high e iciency o ene gy ans e in pho osyn he ic sys ems [10–12]. This is especially no able since i akes place in a scena io appa en ly no p opi ious o he obse a ion o quan um e ec s. In pa icula , i has been sugges ed ha high e iciency anspo a ises as a esul o he dynamical in e play be ween he quan um cohe en e olu ion o he pho osyn he ic sys em and he dephasing noise in oduced by i s su ounding en i on- men , a phenomenon called en i onmen -assis ed quan um anspo (ENAQT) [13] o dephasing-assis ed ene gy anspo [14]. As s a ed in Re . [15], ENAQT can be unde s ood as he supp ession o cohe en quan um localiza ion media ed by noise, helping he exci a ion o mo e as e h ough he pho osyn he ic sys em, hus inc easing he e iciency o ene gy anspo . In his way, ENAQT migh be seen as a phenomenon ha exis s only in a egime whe e he quan- um and classical wo lds o e lap. No wi hs anding, making use o he quan um-classical co espondence o elec onic ene gy ans e p esen ed in Re . [16], we show he e ha he same e ec can also be ound in pu ely classical sys ems. Ou depa u e poin is based on he conside a ion ha agg ega es o coupled monome s (such as he FMO complex) can also be desc ibed as a sys em o weakly in e ac ing classical oscilla o s [17]. We hen demons a e ha he noise-assis ed enhancemen o anspo e iciency in he FMO complex, shown in Re s. [13,14], and based on a pu e quan um o malism, can also be ound in a pu ely classical model, wi hou he need o eso o quan um e ec s. Fo he sake o compa ison and cla i y, we will i s model he FMO complex as a quan um sys em o N in e ac ing si es, whe e he in e ac ion o each si e wi h i s su ounding en i onmen is modeled by a pu e dephas- ing p ocess. We ha e adop ed his model because o i s ex ended use o desc ibing noise-assis ed ene gy ans e p ocesses in pho osyn he ic sys ems [13,14]. Nex , we will p esen he classical model o Re s. [16,17], which co esponds o a sys em o Nweakly coupled ha monic classical oscilla o s. In his case, en i onmen al e ec s a e in oduced by assuming ha he equency o each oscil- la o a ies s ochas ically as a Gaussian Ma ko p ocess. Finally, we will sol e bo h models using he si e ene gies and coupling coe icien s o he FMO complex o P os hecochlo is aes ua ii o show ha he same en i onmen -assis ed ene gy ans e e ec can be ound in bo h classical and quan um models. The Hamil onian o a sys em comp ising Nin e ac ing si es in he p esence o a single exci a ion is gi en by PRL 110, 218101 (2013) PHYSICAL REVIEW LETTERS week ending 24 MAY 2013 0031-9007=13=110(21)=218101(4) 218101-1 Ó2013 Ame ican Physical Socie y ^ HS¼X N n¼1 njnihnjþX N nÞm Vnmjnihmj;(1) whe e jnideno es he exci a ion being a si e n. The n h-si e ene gies and he coupling be ween si es nand m a e desc ibed by nand Vnm, espec i ely. We make use o a simple model whe e he dynamics o he sys em in e ac ing wi h a su ounding en i onmen is desc ibed by a Lindblad mas e equa ion, which in he Bo n-Ma ko and secula app oxima ions is w i en as [18] @^nm @ ¼i @½^ HS;^nm þ^ Ldeph½^nm þ^ D½^nm:(2) He e, he in e ac ion o he sys em wi h he en i onmen is cha ac e ized by a pu e dephasing p ocess gi en by he Lindblad ope a o ^ Ldeph½^nm ¼½1=2ðnþmÞ ffiffiffiffiffiffiffiffiffiffiffiffi nm pnm^nm, wi h nbeing he dephasing a es. Al hough he pu e dephasing model is no able o cap u e impo an aspec s o elec onic ene gy ans e , such as phonon elaxa ion [19], i p o ides a use ul desc ip ion o en i onmen al e ec s in a simple way. To quan i y he ans e o ene gy om a chosen si e k o he eac ion cen e , we ha e phenomenologically in oduced an i e- e sible decay p ocess (wi h a e ) desc ibed by he ope a o ^ D, which is gi en by [20]^ D½^nm ¼ jki hkj;^gnm, whe e g s ands o he an icommu a o . Making use o Eq. (2), one can de ine a measu e o he e iciency o ene gy anspo as he popula ion ans e ed o he eac ion cen e , wi hin a ime ,as Qe ¼2 Z 0hkj^ðsÞjkids: (3) Equa ions (2) and (3) cons i u e he quan um equa ions, which ha e o be compa ed wi h he equa ions ha will be ob ained in he classical model. Fo he classical case, we conside an ensemble o N coupled ha monic oscilla o s, each wi h mass Mand e- quency !n. The empo al e olu ion o he sys em is desc ibed by a classical Hamil onian, which in e ms o he posi ion qnand momen um pno each oscilla o eads as HS¼X np2 n 2MþM!2 n 2q2 nþ1 2X nÞm Knmqnqm;(4) whe e Knm s ands o he coupling coe icien be ween he oscilla o s. By de ining a new dimensionless com- plex ampli ude [21], ~znð Þ¼~ qnð Þþi~ pnð Þ, wi h ~ qn¼ ðM!n=2@Þ1=2qnand ~ pn¼ð2@M!nÞ1=2pn, he Hamil on equa ions o mo ion o he sys em can be cas in o a single equa ion: @~zn @ ¼i!n~zniX m ~ Knm Re ~zmg:(5) Re g s ands o he eal pa o a complex numbe and ~ Knm ¼Knm=ðMffiffiffiffiffiffiffiffiffiffiffiffiffi !n!m pÞ. To include en i onmen al e ec s, we p oceed in he same manne as in he cons uc ion o a Kubo oscilla o [22,23]. Fo his, we assume ha he equency o each classical oscilla o a ies andomly as a s ochas ic p ocess: !nð Þ¼!nþnð Þ.!nis now he a e age equency o he n h oscilla o and nð Þdesc ibes a Gaussian Ma ko p ocess wi h ze o a e age (Wiene p ocess), i.e., hnð Þi ¼ 0and hnð Þmð 0Þi ¼ nnmð  0Þ, whe e hi deno es s ochas ic a e aging. In Re . [16], i has been shown ha one can ans o m Eq. (5), wi hin he amewo k o I o ˆcalculus [24], in o a classical mas e equa ion ha desc ibes he empo al dy- namics o he sys em o coupled ha monic oscilla o s, when in e ac ion wi h he su ounding en i onmen is aken in o accoun . To desc ibe he ans e o exci a ion om he k h oscilla o o he eac ion cen e , we ex end his esul , and in oduce an i e e sible decay p ocess (wi h a e ), desc ibed by D½nm ¼ jkihkj;gnm, whe e nm ¼h~zn~z mi. In his way, we can w i e he classical mas e equa ion as @nm @ ¼H½nm þL½nm þD½nm þi @X jðVmjh~zj~zniVnjh~z j~z miÞ;(6) wi h Vnm ¼~ Knm@=2, and H½nm ¼ið!n!mÞnm i @X jðVnjjm VjmnjÞ;(7) L½nm ¼ 1 2ðnþmÞ ffiffiffiffiffiffiffiffiffiffiffiffi nm pnmnm:(8) The ene gy ans e e iciency wi hin he ensemble o oscilla o s is hen gi en by Ce ¼2 Z 0 kkðsÞds: (9) ð Þ¼ð Þ=Pnnn is he no malized classical densi y ope a o . Equa ions (6) and (9) ep esen he classical equa ions, whose esul s ha e o be compa ed wi h hei quan um- mechanical coun e pa , Eqs. (2) and (3). To his end, we can make use o he si e ene gies and coupling coe icien s o he FMO complex o P. aes ua ii [25]. The FMO is a pigmen -p o ein complex ha guides he ene gy om he ligh -ha es ing chlo osomes o he eac ion cen e in g een sul u bac e ia [26,27]. I is a ime o h ee iden ical subuni s in e ac ing weakly wi h each o he . Each subuni is composed o se en bac e iochlo ophyll-a (BChla) mole- cules embedded in a sca olding o p o ein molecules, as shown in Fig. 1. The FMO complex is gene ally modeled PRL 110, 218101 (2013) PHYSICAL REVIEW LETTERS week ending 24 MAY 2013 218101-2 by a ne wo k o se en di e en si es, whe e he dynamics o a single exci a ion h ough he complex is go e ned by he speci ic alues o he si e ene gies (n) and he coupling coe icien s (Vnm). In pa icula , we will use he alues o he si e ene gies and coupling coe icien s o P. aes ua ii, as s a ed in Tables 2 and 4 o Re . [25]. The ini ial s a e o he sys em co esponds o a single exci a ion in si e 1. In he FMO, he BChl 3 is in he icini y o he eac ion cen e [25]. Thus, we ake his si e (k¼3) as he main exci a ion dono o he eac ion cen e , wi h a ans e a e es ima ed o be ¼1ps 1[28]. Fu he mo e, o he pu e dephasing p ocess, we conside ha dephasing a es a e he same o all si es (¼n) and ha he e iciency o ene gy ans e is limi ed by he ini e exci a ion li e ime ( 1ns). Figu e 2shows he e iciency o ene gy ans e as a unc ion o he dephasing a e ob ained by means o he quan um equa ions (2) and (3). No ice ha a low dephas- ing, i.e., wi h en i onmen e ec s no conside ed, cohe en e olu ion o he sys em leads o an e iciency o abou 90%. When inc easing he dephasing, e iciency g ows o almos 100%, showing ha he en i onmen a ec s he sys em in such a way ha i becomes mo e e icien o ans e ing ene gy o he eac ion cen e . Finally, o s onge dephas- ing, e iciency d ops apidly and almos no ene gy is ans- e ed o he eac ion cen e . Quali a i ely simila esul s ha e also been ob ained o he case o he FMO complex o Chlo obium epidum [13]. We now u n ou a en ion o he case o he classical model by sol ing Eqs. (6) and (9). Figu e 3shows he e iciency o he ene gy ans e as a unc ion o he dephas- ing a e. We obse e ha he same noise-assis ed e ec is also p esen in he pu ely classical model. Fo he sake o compa ison, Fig. 3also shows he solu ion o he quan um- mechanical model (dashed line). No ice ha bo h solu ions ag ee o dephasing a es up o 103ps1. Howe e , o la ge alues o dephasing he quan um and classical solu- ions di e om each o he . This is in ag eemen wi h he ac ha bo h solu ions a e he same, p o ided ha he condi ion !nis sa is ied [16]. Noise-assis ed ene gy anspo in diso de ed sys ems has been unde s ood as he supp ession o cohe en quan- um localiza ion h ough noise, b inging he de uned quan- um le els in o esonance and hus acili a ing he ene gy ans e [13,15]. No wi hs anding, he esul s p esen ed he e show ha he same e ec can also be ound in pu ely classical sys ems. This implies ha one can make use o such sys ems in o de o simula e he in ica e ene gy ans e mechanisms ha ake place in molecula agg e- ga es, such as he pho osyn he ic FMO complex. Recen ly, i has been sugges ed ha classical LC ci cui oscilla o s (whe e Ls ands o induc ance and C o capaci ance) can be used o model coupled quan um wo- le el sys ems [29]. Hence, one could de ise an expe imen- al appa a us comp ising eigh elec ical oscilla o s wi h he eigh h ac ing as he eac ion cen e , which would be FIG. 2 (colo online). Ene gy ans e e iciency as a unc ion o he dephasing a e ob ained om he quan um-mechanical equa ions. FIG. 3 (colo online). Ene gy ans e e iciency as a unc ion o he dephasing a e ob ained om he classical equa ions (solid line). Fo he sake o compa ison, we ha e also included he e he cu e shown in Fig. 2, which co esponds o he solu ion o he quan um equa ions (dashed line). FIG. 1 (colo online). A angemen o he BChla molecules o a single uni o he Fenna-Ma hews-Olson (FMO) complex. The igu e was c ea ed using PyMOL [35], and is based on he P o ein Da a Bank en y 3ENI. PRL 110, 218101 (2013) PHYSICAL REVIEW LETTERS week ending 24 MAY 2013 218101-3 s ongly coupled o one o he emaining oscilla o s. Then, by s ochas ically modula ing he equencies !n, and p op- e ly con olling he noise in ensi y n, one would be able o obse e he noise-assis ed ene gy ans e phenomenon by moni o ing he signal p esen in he eigh h oscilla o . These classical simula ions could be u he used o compa e wi h he ecen expe imen al p oposal o noise-assis ed ans- po based on coupled quan um-op ical ca i ies [30]. The concep o noise-assis ed ene gy anspo has been ex ensi ely used o desc ibing he inne wo king o quan- um and classical sys ems [31]. Along hese lines, he pa icula enhancemen e ec desc ibed in his Le e migh open a new esea ch di ec ion owa ds new me hods o enhancing he e iciency o a my iad o ene gy ans- po sys ems ha ine i ably li e in a noisy en i onmen , om small-scale in o ma ion and ene gy ans e sys ems in mic owa e and pho onic ci cui s o long-dis ance high- ol age elec ical lines. In his way, a speci ic ea u e ini ially concei ed in a quan um scena io (en i onmen - assis ed ene gy anspo ) is shown o a ise as well in a pu ely classical con ex , widening hus he scope o pos- sible quan um-inspi ed echnological applica ions. 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