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.
To conclude, he sea ch and demons a ion o sys ems in
which o obse e quan um-mechanical e ec s wi h no
classical coun e pa is a subjec o li ely in e es and
deba e [32–34]. Biological sys ems a e no , in p inciple,
a p opi ious scena io o he obse a ion o quan um ea-
u es, such as quan um supe posi ion, in e e ence, o
en anglemen . Ne e heless, 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. This does
no mean, howe e , ha in ce ain cases a pu ely classical
model may no simila ly ep oduce some o he esul s
p edic ed by he ull quan um-mechanical model, since
classical physics eme ge, a e all, om quan um physics
unde many ci cums ances.
This wo k was suppo ed by p ojec s FIS2010-14831
and FET-Open G an No. 255914 (PHORBITECH). This
wo k has also been pa ially suppo ed by Fundacio
P i ada Cellex Ba celona.
*[email p o ec ed]
†
[email p o ec ed]
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218101-4