Exciton-phonon information flow in the energy transfer process of photosynthetic complexes

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by  Patrick Rebentrost

Exciton-phononinformationflowintheenergytransferprocessofphotosyntheticcomplexes
Patrick Rebentrost1,∗ and Ala´n Aspuru-Guzik1,†
1DepartmentofChemistryandChemicalBiology,HarvardUniversity,12OxfordSt.,Cambridge,MA02138
(Dated: January25,2011)
Non-Markovianandnon-equilibriumphononeffectsarebelievedtobekeyingredientsintheenergytransfer
inphotosyntheticcomplexes,especiallyincomplexeswhichexhibitaregimeofintermediateexciton-phonon
coupling. Inthiswork,weutilizearecently-developedmeasurefornon-Markovianitytoelucidatetheexciton-
phonondynamicsintermsoftheinformationflowbetweenelectronicandvibrationaldegreesoffreedom. We
studythemeasureinthehierarchicalequationofmotionapproachwhichcapturesstrongsystem-bathcoupling
effectsandnon-equilibriummolecularreorganization.Weproposeanadditionaltrace-distancemeasureforthe
informationflowthatcouldbeextendedtoothermasterequations. Wefindthatforamodeldimersystemand
theFenna-Matthews-Olsoncomplexthatnon-Markovianityissignificantunderphysiologicalconditions.
1
1 The initial step in photosynthesis involves highly efficient answerishowmuchinformationisexchangedbetweenexci-
0 excitonic transport of the energy captured from photons to a tonic and phononic degrees of freedom in that process and,
2
reactioncenter[1]. Inmosthigherplantsandotherorganisms specifically,howmuchinformationreturnsfromthephonons

n suchasgreensulphurandpurplebacteriathisprocessoccurs totheexciton.
a inlight-harvestingcomplexeswhichconsistofelectronically Measurefornon-Markovianity−Arecentareaofresearch
J
coupled chlorophyll molecules embedded in a solvent and a isthestudyofmeasurestocharacterizenon-Markovianity[13,

4
proteinenvironment[2].Severalrecentexperimentsshowthat 14]. We utilize the readily applicable measure developed by
2
excitonic coherence can persist for several hundreds of fem- Breuerandco-workers[13]. Itisbasedonthequantumstate

tosecondsevenatphysiologicaltemperature[3].Theseexper- tracedistance:
]
h imentssuggestthehypothesisthatquantumcoherencemaybe
1
p biologicallyrelevantforphotosynthesis. D(ρ (t),ρ (t))= Tr{|ρ (t)−ρ (t)|}, (1)
- 1 2 2 1 2
t The challenging regime of intermediate coupling of the √
n
electronictothevibrationaldegreesoffreedommotivatesthe where|A|= A†A.Thismeasurequantifiesthedistinguish-
a
study of sophisticated master equations with non-Markovian ablility of two quantum states ρ and ρ . For a completely
u 1 2
q effects. Several approaches have been taken, ranging from a positivetrace-preservingmapE thetracedistanceiscontrac-
[ polarontransformation[4], quantumstatediffusion[5], gen- tive, i.e. D(E(ρ ),E(ρ )) ≤ D(ρ ,ρ ) [15]. Time in-
1 2 1 2
eralized Bloch-Redfield [6], non-Markovian quantum jumps tervals in which D increases indicate non-Markovian infor-
2
[7,8],quantumpathintegrals(QUAPI)[9],todensitymatrix mationflow[13], inourcasefromthevibrationaldegreesof
v
9 renormalization group (DMRG) [10]. After photoexcitation, freedombacktotheelectronicdegreesoffreedom.Intermsof
0 thenuclearcoordinatesofamoleculewillrelaxtonewequi- theslopeofD,i.e. σ(ρ (t),ρ (t)) = dD(ρ (t),ρ (t)),
1 2 dt 1 2
8 libriumpositions,aphenomenonthatisborneoutbythetime- this implies that σ > 0 for these time intervals. Formally,
3 dependentStokesshift,thatisthefrequencyshiftbetweenab- a measure for non-Markovianity is defined by the following
1. sorption and emission spectra. Redfield theory assumes that [13]:
1 thephononbathisalwaysinthermalequilibriumandthuscan- (cid:90) ∞
0 not capture this effect [11]. Ishizaki and Fleming employed NM-ity= max dtI(σ(t)) σ(ρ (t),ρ (t)).
1 the hierarchical equation of motion (HEOM) approach [12], ρ1(0),ρ2(0) 0 1 2
v: which interpolates between the usual weak and strong (sin- (2)
i gular)exciton-phononcouplinglimitsandtakesintoaccount Here, the indicator function I(x>0) = 1 (I(x<0)=0)
X non-equilibriummolecularreorganizationeffects. monitors only the increasing part of the trace distance time
r Inthiswork,westudythenon-Markovianityoftheexciton evolutioni.e. theinformationbackflow. Theoptimizationof
a
transferprocessbymeansofnumericalsimulation. Quantum theinitialstatesinEq.(2)obtainsthemaximallypossiblenon-
mechanical time evolution under decoherence leads to trans- Markovianityofaparticularquantumevolution.Inthecaseof
fer of information encoded in the electronic excited state to photosynthetic energy transfer, initial states are given by the
the vibrational degrees of freedom. Non-Markovianity can particularphysicalsituationandleadtosmallervaluesforthe
becharacterizedasthereturnofthisinformationtotheelec- non-Markovianity.Weusebothphysicalandoptimizedinitial
tronic degrees of freedom. To quantify the exciton-phonon statesinthiswork.
informationflow,weemploythestate-of-theartmasterequa- Hierarchy equation of motion− The Hamiltonian
tionapproachbyIshizakiandFlemingandanewly-developed describing a single exciton in a complex with N
measure for non-Markovianity. The central question we will (cid:80)molecules is given by He = (cid:80)Nm=1((cid:15)m + λ)|m(cid:105)(cid:104)m| +
J (|m(cid:105)(cid:104)n|+|n(cid:105)(cid:104)m|). The site energies (cid:15) , and
m<n mn m
couplingsJ areobtainedfromdetailedquantumchemistry
mn
studies and/or fitting of experimental data. The set of states
∗[email protected] |m(cid:105)denotesthesitebasis. Thereorganizationenergyλisthe
†[email protected] energy difference of the non-equilibrium phonon state after
2
Franck-Condon excitation and the equilibrium phonon state 1.0
a
andisassumedtobethesameforeachsite. Thesuperopera- e 0.8
c
tor corresponding to H is L ρ = [H ,ρ]. The Hamiltonian n
for the phonon environement ies H =e(cid:80) (cid:126)ω (cid:0)p2+q2(cid:1)/2, sta 0.6
ph i i i i di
where ωi, pi, and qi are the frequency, dimensionless mo- ce 0.4 (cid:76)
mentum and position operators of the phonon mode i. The Tra 0.2
couplingoftheelectronicdegreesoffreedomtothephonons
(cid:80) (cid:80) 0.0
isgivenbytheHamiltonianH = V ( g q ) ,
ex−ph m m i i i m 0 200 400 600 800 1000
with V = |m(cid:105)(cid:104)m| and where we assume that site-energy
m
fluctuationsdominate,thebathdegreesoffreedomateachsite Time fs
areuncorrelated,andthecouplingconstantstotherespective
e
modes are given by the constants gi. The spectral density, anc 2.0 b
wphhoicnhondemscordibeess,thisecgoivuepnlinbgystJre(nωg)tho=fex2cλitγoωn/to(cid:0)ωp2ar+ticγu2la(cid:1)r. dist 1.5 (cid:64) (cid:68)
O
Perhapsthemostrelevantparameter,thebathdissipationrate D 1.0 (cid:76)
A
γ, is related to the bath correlation time by τc = 1/γ. The al 0.5
equation of motion for the reduced single-exciton density ot
T 0.0
matrix ρ is obtained by tracing out the less relevant phonon
0 200 400 600 800 1000
degrees of freedom and utilizing Wick’s theorem for the
Gaussianfluctuations,leadingtoanon-perturbativehierarchy Time fs
ofsystemandauxilliarydensityoperators(ADOs)[12]:
∂ (cid:32) (cid:88)N (cid:33) FIG.1. (a)Tracedistanceofthestatesρ1 =|1(cid:105)(cid:104)1|andρ2 =|2(cid:105)(cid:104)2|
∂tσn(t)=− iLe+ nmγ σn(t) (3) aesquaatfiuonnctoiofnmooftitoimnemfoodreal.twToh-esittreacsyes(cid:64)dteism(cid:68)tan(dciemeevro)livnesthferohmiermaracxhiy-
m=1
mallydistinguishable(=1)fortheinitialstatestoindistinguishable
N N
(cid:88) (cid:88) (= 0)forthethermalequilibriumstate. Atintermediatetimes, the
+ φ σnm+1(t)+ n θ σnm−1(t).
m m m trace distance increases, which is due to a reversal of the informa-
m=1 m=1 tion flow. Various bath correlation times are shown, 150 fs (blue),
The system density matrix is the first member of the hierar- 100fs(red),and50fs(yellow). Forlongcorrelationtimes,thatis,
chy ρ(t) = σ0(t). The other members of the hierarchy are a bath with more memory, the regions of increasing trace distance
arranged in tiers and indexed by n = (n1,.....,nN) ≥ 0, aremorepronounced. (b)TheADOdistanceDAtoDtO illustratesthe
where a single tier is given by all elements where (cid:80) n informationcontentinthenon-Markoviandegreesoffreedomofthis
m m modelforthesamesystem,initialstates,andbathcorrelationtimes
is constant. The notation n = (n ...,n ±1,...,n )
m±1 1, m N as in (a). The anticorrelated oscillations of the two figures clearly
is introduced. The superoperators in the high temperature
showtheinformationflowbetweenthesystemandtheNMdegrees
regime (cid:126)γβ < 1 are φ σn = i[V ,σn] and θ σn =
i(cid:0)2λ/β(cid:126)2[V ,σn]−iλγm/(cid:126){V ,σn}(cid:1)mand act onmsystem offreedom.
m m
andADOs,where{,}istheanticommutator. Initially,allthe
hierarchy members are zero, which corresponds to the elec- describeperturbationsonthetime-evolvedexcitonicstatedue
tronicgroundstatephononequilibriumconfiguration.Fornu- to the phonon environment. This separation into system and
merical propagation, the hierarchy is truncated based on the auxiliary degrees of freedom implies a quantification of the
criterion (cid:80)mnm (cid:29) ωγe, where ωe is a characteristic fre- information content of the auxiliary systems at a given point
quencyofL . Weassumethatthisconditionissatisfiedwhen intime:
e
theT(cid:80)hemrnepmreissebnytaatiofanctoorf5tlhaergehriethraarnchωγye.equation of mo- DAtoDtO(t)= (cid:88)D(σ˜n1 (t),σ˜n2 (t)). (4)
tion in Eq. (3) is not unique. It can be advan- n(cid:54)=0
tageous to expand the hierarchy in a set of normal-
Thismeasuredependsontheelectronicinitialstatesand,im-
ized auxiliary systems [16]. Redefining the ADOs as
portantly, uses the normalized ADOs. Although the σ˜n(t)
σ˜n(t) = ((cid:81)mnm|c0|−nm)−1/2σn(t) and rescaling the are not quantum states for n (cid:54)= 0, the trace distance never-
superoperators as φmσnm+1 = (cid:112)(nm+1)|c0|φmσ˜nm+1 theless measures differences of these operators arising from
and nmθmσnm−1 = (cid:112)nm/|c0|θmσ˜nm−1, with c0 = thedifferentinitialstates. Thismeasurecouldinprinciplebe
(cid:0)2λ/β(cid:126)2−iλγ/(cid:126)(cid:1) in the high temperature regime, leads to extended to other families of master equations that employ
amorebalancedEOMandcanleadtoalowerrequirementin auxiliarydegreesoffreedom.
termsofthenumberofADOs. Therescaledrepresentationis Results for a dimer− In this section, a two-molecule sys-
usefulforsimulatinglargesystemssuchastheFMOcomplex tem, or dimer, is studied in terms of the measure for non-
andforstudyingtheADOsthemselves,asfollows. Markovianity, using, if not otherwise mentioned, the regu-
Tocharacterizetheinformationflowfromthepointofview lar hierarchy equation of motion approach Eq. (3) and scan-
ofthenon-Markoviandegreesoffreedom,weproposeanother ning over the parameters of the model. As standard pa-
measure based on the trace distance. The auxiliary systems rameters, we use the site energies (cid:15) = 0, (cid:15) = 120/cm,
1 2
3
astnrdontghley-ccoouupplliendgbJact=erio−c8h7lo.7ro/pcmhy.llTshiitse c1oarrnedsp2onsudbsstyosttehme vianity 000...233505 a 0.15 b
of the Fenna-Matthews-Olson complex. Along the lines of ko 0.20 0.10
ar 0.15
the discussion in [12], the standard bath correlation time is M 0.10 (cid:76) 0.05
taken from 50 fs to 150 fs. We assume ambient temperature n(cid:45) 0.05 (cid:76)
No 0.00 0.00
T = 288 K (200/cm). The main results use 40 tiers (819 20 40 60 80100120140 0 50 100150200250300
ADOs),andEq. (3)isintegrateduptomaximally20ps.
Coupling 1cm Siteenergydifference 1cm
First, in Fig. 1 (a), we investigate the time evolution of
y
the trace distance in Eq. (1) for the dimer system at reor- nit 0.6 c 0.20 d
ganization energy λ = 20/cm. We choose the two initial via 0.5 (cid:64) (cid:144) (cid:68) 0.15 (cid:64) (cid:144) (cid:68)
o 0.4
k
statesρ1 = |1(cid:105)(cid:104)1|andρ2 = |2(cid:105)(cid:104)2|andthestandardparame- Mar 00..23 (cid:76) 0.10 (cid:76)
tersgivenabove,whilevaryingthebathdissipationrate. One n(cid:45) 0.1 0.05
can see clear non-Markovian revivals in the time intervals at No 0.0 0.00
20 40 60 80 100120140 0.1 1 10 100
around 150 fs and around 300 fs, borne out by increases in
the trace distance. For larger correlation time of the bath of Bathdissipationrate 1cm Reorganizationenergy 1cm
cdrτeiictlsaIas=nttiiipooFa1nnitg5eit.0dsim1mfasew(obtτarh)eyc,ewls=rieakepe5riieln0dyvvlfityevso,s.atarlisesgtaauotrrceenctumhtoreostrthiienmeptserhyoeesnvtcoeoamulsuneticinooesdnft.esoamAfdathobllefaetbrAhecDieonxOrg-- NonMarkovianity(cid:45) 00000.....02468(cid:42)e(cid:42)(cid:76)(cid:42)(cid:42)F(cid:42)M(cid:42)(cid:42)O(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:64)(cid:42)(cid:42)(cid:144)(cid:42)(cid:42)(cid:42)(cid:42)(cid:68) 000000......001122050505(cid:42)(cid:42)(cid:42)f(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:76)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)F(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)M(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)O(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:64)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:144)(cid:42)(cid:42)(cid:42)(cid:42)(cid:68)
distance measure in Eq. (4) for the same system, using the 20 40 60 80 100120140 0 20 40 60 80 100120
normalizedauxiliarysystems. Atsmalltimes, theADOsare Bathdissipationrate 1cm Reorganizationenergy 1cm
all zero, thus Dtot = 0, while at long times the system
ADO
andADOsconvergetothethermalequilibriumandallinfor- FIG.2. Non-Markovianityasafunctionoftheparametersofadimer
mation about the initial states is dissipated. At intermediate system,a)-d),andtheFenn(cid:64)a(cid:144)-Ma(cid:68)tthews-Olsoncomplex,e)and(cid:64) f(cid:144)),i(cid:68)n
times,onefindsstronginformationflowbetweensystemand thehierarchyequationofmotionapproach. Forthedimer: a)cou-
thenon-Markoviandegreesoffreedom,whichisborneoutby plingJ12,b)siteenergydifference(cid:15)2−(cid:15)1,c)bathdissipationrate
γ(whichisrelatedtothebathcorrelationtimebyγ =1/τ ,)andd)
theanticorrelatedoscillationsin(a)and(b),respectively. For c
reorganizationenergyλ. Thesolidlinesarecalculationsperformed
this, notethe dips at around 150 fs and around 300 fs in (b).
withtheinitialstatesbeingρ = |1(cid:105)(cid:104)1|andρ = |2(cid:105)(cid:104)2|whilethe
Additionally,theinformationcontentintheADOsissmaller 1 2
dots are calculations performed with optimized initial states. The
whenthebathdissipationrateislarger.
dashedpartofthelinesindicatesthataparameterwasscannedbe-
The computation of the non-Markovianity measure in Eq. yondthevalidityoftheequationbasedonthetruncationcondition.
(2) involves an optimization of the initial states. On the one FortheFMOcomplex:e)bathdissipationrateandf)reorganization
hand, the following results use the two initial states ρ = energy. Inallfigures,exceptinc)ande),weusethedifferentbath
1
|1(cid:105)(cid:104)1| and ρ = |2(cid:105)(cid:104)2| as above, but one the other hand, correlationtimes:50fs(blue),100fs(purple),and150fs(red).
2
compare to a systematically-optimized pair of initial states.
Oneinitialstateischosenoutof50purestatesontheBloch
sphere.Thisleadstoatotalnumberof1225independentpairs sure as a function of the inverse bath correlation time, i.e.
ofstatesintheoptimization. (Thestateoptimizationitselfis the bath dissipation rate. A longer τ leads to more non-
c
performedwith20tiersofADOs.) Markovianity, because a sluggish bath keeps the memory of
We now proceed to scan the NM-ity measure over the pa- excitations much longer and the information is able to re-
rameters of the model, beginning with the system parame- turn back to the electronic degrees of freedom. Second, the
ters,thesiteenergydifferenceandthechromophoriccoupling. othersignificantbathparameter,thereorganizationenergy,is
First, theNM-ityisstronglydependentonthechromophoric ameasureofthenon-equilibriumcharacterofthephononbath
coupling,seeFig. 2(a). Atzerocoupling,themoleculesare intheexcitedstate. InFig. 2(d), weshowtheNMmeasure
independentandtheNM-ityiszero; thetwoinitialstatesre- as a function of the reorganization energy. At zero λ, which
mainperfectlydistinguishable. Atnon-zerocoupling,thedy- essentiallymeansnocouplingtothephonons,theunitarydy-
namicsshowsincreasingNMrevivals,whilebothinitialstates namicsobviouslyshowszeronon-Markovianity. Atweakλ,
convergetothesamethermalequilibriumstate. Second,with we observe a NM-ity of around 0.1 at t = 150 fs, which is
c
respect to the site energy difference, the dependence of the completelyneglectedbyRedfieldtheory.Atintermediateλof
NM-ity weakly slopes downward, see Fig 2 (b). As the en- around40/cm,closetothephysiologicalvaluesoftheFenna-
ergy difference increases the coupling becomes less signifi- Matthews-Olson complex, the NM-ity is maximal, showing
cantandthemoleculesbecomemoreindependent. Thisleads a value of around 0.2. At large λ, the regime of incoherent
todecreasedNM-ity. transport[12],theNM-ityvanishes.
Next,weinvestigatetheroleofthebathparameters. First, Results for the Fenna-Matthews-Olson complex− We now
as mentioned before, the bath correlation time crucially af- investigatetheexciton-phononinformationflowintheFenna-
fects the memory of the bath and the information back flow Matthews-Olson complex. The FMO complex is found in
to the system. In Fig. 2 (c), we show the NM-ity mea- greensulphurbacteria,whereitconnectstheantennacomplex
4
with the reaction center during the energy transfer process. [13,14],canleadtoagreaterunderstandingoftheexcitondy-
TheFMOcomplexisatrimerwithsevenbacteriochlorophyll namics in these systems. In principle, the recently proposed
(BChl) molecules in each subunit and with one additional ultrafastquantumprocesstomographyprotocolextractsafull
BChl molecule shared between the units. We implement the characterization of the quantum map [18], and could thus be
hierarchy equation of motion for the seven sites (N = 7), usedfortheexperimentalsideofthistask. However, experi-
using the method for rescaling the auxilliary systems as dis- mentsspecificallydesignedtoextractaparticularobservable,
cussed above and in [16]. As was shown recently [17], with suchasinourcasetheNM-itymeasure,couldbecarriedout
fourtiers(329ADOs)thecoherentpopulationdynamicscan withareducednumberofexperimentalrequirements. Future
be accurately simulated, but due to the reduced number of workwillinvestigateanalyticallyandnumericallyhowtoeffi-
ADOs,weexpecttominimallyoverestimatetheNM-ity. We cientlyextractthenon-Markovianityfromthephase-matched
use the initial states ρ = |1(cid:105)(cid:104)1| and ρ = |2(cid:105)(cid:104)2|. First, the signalobservedinfour-wavemixingexperiments.
1 2
scan of γ obtains a strong dependence similar to the dimer To summarize, we have shown numerically that there is a
system,seeFig. 2(e). Second,thescanofthereorganization considerable non-Markovian information flow between elec-
energy obtains a maximum at around λ = 55/cm with the tronic and phononic degrees of freedom of light-harvesting
NM-ity being 0.2 at λ = 35/cm and τ = 150 fs, see Fig. chromophoricsystemsunderphysiologicalconditions,utiliz-
c
2 (f). The results suggest that non-Markovian effects should ingarealisticmodelforsystemssuchastheFenna-Matthews-
playasignificantroleinthefunctionoftheFenna-Matthews- Olson complex. The results suggest that non-Markovianity
Olsoncomplex. playsasignificantroleinphotosyntheticenergytransfer.
The authors are grateful to J. Piilo, C. Rodriguez-Rosario,
Conclusion− The theoretical and experimental characteri- S. Saikin, and J. Zhu for insightful discussions. The authors
zation of photosynthetic light-harvesting complexes in terms acknowledgefundingfromDOE,DARPA,andcomputational
of non-Markovianity, as e.g. defined in the recent work resourcesthroughHarvardResearchComputing.
[1] R. E. Blankenship, Molecular Mechanisms of Photosynthesis, ChemicalPhysicsLetters478,234(2009).
1sted.(Wiley-Blackwell,2002)p.336. [10] J.Prior,A.Chin,S.Huelga, andM.Plenio,PhysicalReview
[2] Y.-C. Cheng and G. R. Fleming, Annual Review of Physical Letters105,1(2010).
Chemistry60,241(2009). [11] A.IshizakiandG.R.Fleming,TheJournalofChemicalPhysics
[3] G.S.Engel,T.R.Calhoun,E.L.Read,T.-K.Ahn,T.Mancal, 130,234110(2009).
Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, Nature [12] A.IshizakiandG.R.Fleming,TheJournalofChemicalPhysics
446,782(2007);H.Lee,Y.-C.Cheng, andG.R.Fleming,Sci- 130,234111(2009); ProceedingsoftheNationalAcademyof
ence316,1462(2007); G.Panitchayangkoon,D.Hayes,K.A. SciencesoftheUnitedStatesofAmerica106,17255(2009).
Fransted,J.R.Caram,E.Harel,J.Wen,R.E.Blankenship, and [13] H.-P.Breuer,E.-M.Laine, andJ.Piilo,PhysicalReviewLetters
G.S.Engel,ProceedingsoftheNationalAcademyofSciences 103,1(2009).
oftheUnitedStatesofAmerica107,1(2010);E.Collini,C.Y. [14] M. Wolf, J. Eisert, T. Cubitt, and J. Cirac, Physical Review
Wong,K.E.Wilk,P.M.G.Curmi,P.Brumer, andG.D.Sc- Letters 101, 1 (2008); A. K. Rajagopal, A. R. U. Devi, and
holes,Nature463,644(2010). R. W. Rendell, http://arxiv.org/abs/1007.4498 (2010); A. Ri-
[4] S.Jang, Y.-C.Cheng, D.R.Reichman, andJ.D.Eaves,The vas,S.Huelga, andM.Plenio,PhysicalReviewLetters105,1
JournalofChemicalPhysics129,101104(2008). (2010).
[5] J.Roden,A.Eisfeld,W.Wolff, andW.Strunz,PhysicalReview [15] M. A. Nielsen and I. L. Chuang, Quantum Computation and
Letters103,3(2009). Quantum Information, 1st ed. (Cambridge University Press,
[6] J. Wu, F. Liu, Y. Shen, J. Cao, and R. J. Silbey, 2000).
http://arxiv.org/abs/1008.2236 (2010). [16] Q.Shi,L.Chen,G.Nan,R.-X.Xu, andY.Yan,TheJournalof
[7] J. Piilo, S. Maniscalco, K. Ha¨rko¨nen, and K.-A. Suominen, ChemicalPhysics130,084105(2009).
PhysicalReviewLetters100,1(2008). [17] J.Zhu,S.Kais,P.Rebentrost, andA.Aspuru-Guzik,Journal
[8] P.Rebentrost,R.Chakraborty, andA.Aspuru-Guzik,TheJour- ofPhysicalChemistryB(inpress) (2010).
nalofChemicalPhysics131,184102(2009). [18] J. Yuen-Zhou, M. Mohseni, and A. Aspuru-Guzik,
[9] M. Thorwart, J. Eckel, J. Reina, P. Nalbach, and S. Weiss, http://arxiv.org/abs/1006.4866 (2010).