Quantum correlation dynamics in photosynthetic processes assisted by molecular vibrations

1.7 MB·English

by  G. L. Giorgi

Quantum correlation dynamics in photosynthetic processes
assisted by molecular vibrations
G.L.Giorgi
INRIM,StradadelleCacce91,I-10135Torino,Italy
M.Roncaglia
INRIM,StradadelleCacce91,I-10135Torino,Italy
5
F.A.Raffa
1
0 PolitecnicodiTorino,DipartimentodiScienzaApplicataeTecnologia,
2
CorsoDucadegliAbruzzi24,I-10129Torino,Italy

l
u M.Genovese
J
INRIM,StradadelleCacce91,I-10135Torino,Italy
3
1

]
h
p
-
t Abstract
n
a During the long course of evolution, nature has learnt how to exploit quantum effects. In fact,
u recentexperimentsrevealtheexistenceofquantumprocesseswhosecoherenceextendsoverun-
q
expectedlylongtimeandspaceranges.Inparticular,photosyntheticprocessesinlight-harvesting
[
complexesdisplayatypicaloscillatorydynamicsascribedtoquantumcoherence. Here,wecon-

2 sider the simple model where a dimer made of two chromophores is strongly coupled with a
v
quasi-resonant vibrational mode. We observe the occurrence of wide oscillations of genuine
0
quantumcorrelations,betweenelectronicexcitationsandtheenvironment,representedbyvibra-
1
7 tional bosonic modes. Such a quantum dynamics has been unveiled through the calculation of
7 the negativity of entanglement and the discord, indicators widely used in quantum information
0 forquantifyingtheresourcesneededtorealizequantumtechnologies. Wealsodiscussthepos-
1. sibilityofapproximatingadditionalweakly-coupledoff-resonantvibrationalmodes,simulating
0 the disturbances induced by the rest of the environment, by a single vibrational mode. Within
5 this approximation, one can show that the off-resonant bath behaves like a classical source of
1
noise.
:
v Keywords: Quantumeffectsinbiology,Quantumcorrelations,Openquantumsystems
i
X
r
a
1. Introduction
Theexistenceofcoherence,causedbytheinterferenceofprobabilityamplitudeterms,isone
of the distinctive traits of quantum mechanics. Oscillatory behaviours, ubiquitously observed
∗Correspondingauthor
Emailaddress:[email protected](G.L.Giorgi)
PreprintsubmittedtoElsevier July14,2015
in quantum systems, are the consequence of such coherent phenomena. The fact that light-
harvestingcomplexeswereexperimentallyproventoexhibitoscillatoryelectronicdynamicshas
stimulated a deep debate about the nature of such oscillations, and, consequently, the possible
role played by quantum mechanics in biologically functional systems [1, 2, 3, 4, 5]. Clearly,
oscillations can also be found in completely classical systems (the appearance of oscillatory
electronic dynamics within classical models was studied in [6, 7]). Thus, it is of the utmost
importancetogiveaprecisecharacterizationoftheseoscillationsbymeansofsomequantumness
quantifier, in order to unveil the basic mechanisms adopted by nature, which in future could
possiblyinspirenewenergytechnologies.
Thequestionofwhethercoherenceinlight-harvestingcomplexeshasaquantumorclassical
origin was addressed in a series of recent seminal works [8, 9, 10, 11, 12, 13, 14, 15]. In Ref.
[11],Wildeetal.facedtheproblemfromthemacro-realismpointofviewusingtheLeggett-Garg
inequalitiestotestwhetherthesystemdynamicsiscompatiblewithclassicaltheories, whilein
Refs. [9,12,13,14,15,16,17]thepresenceofentanglementamongtheelectronicdegreesof
freedomwasusedtoassessthegenuinequantumcharacterofthewholesystem.
Aslivingobjectsareembeddedintheirownenvironment,thedynamicalbehaviourofelec-
tronicexcitationsisnecessarilyinfluencedbythepresenceofotherdegreesoffreedom,mainly
phonons, whose coupling to the system has been identified to be one of the possible causes
of efficient transport [18, 19, 20, 21, 22, 23]. Stimulated by this observation, in Ref. [24],
O’ReillyandOlaya-Castrorecentlyinvestigatednon-classicalfeaturesofthemolecularmotions
andphononenvironmentsinaprototypedimerthatcanbefoundinlight-harvestingantennaeof
cyanobacteria[25],cryptophytealgae[26,27]andhigherplants[28,29,30]. Acharacterization
ofquantumnesswasperformedbymeansoftheMandelQ-parameterandtheGlauber-Sudarshan
quasi-probabilityPdistribution,whosenegativeregionsinphasespacearenotcompatiblewith
anyclassicaldescriptionofthecoupleddynamics.
Nevertheless, albeit Q-parameter and P-function negativity represent a significant way of
quantifying the quantumness of a system, they do not catch thoroughly the ultimate quantum-
nessrepresentedbynon-classicalityofcorrelations. Forinstance,theP-functionofanEinstein-
Podolski-Rosen(EPR)pairispositive,inspiteofthedeepquantumnatureofthisstate[31].
Inthispaper,wewanttoovercomethisdrawbackbydirectlyquantifyingthedegreeofquan-
tumnessdevelopedduringthecoupledexciton-vibrationdynamicsthroughtheuseofentangle-
ment and of quantum discord between system and environment. On the one hand, entangle-
ment is usually viewed as an utterly fragile property at room temperatures, as it can be easily
destroyed by decoherence [32]. Despite these caveats, however, its presence was predicted in
different biological processes [8, 12, 13, 14, 15, 9, 16, 17]. On the other hand, there can exist
quantumcorrelationsalsointheabsenceofentanglement,aswitnessedbymanyquantumproto-
cols. Thesequantumcorrelationsarecapturedbyquantumdiscord,whosedefinitionoriginates
from two definitions of the classical mutual information whose quantized versions turn out to
be nonequivalent [33]. Quantum discord provides us a criterion of quantumness that is both
necessary and sufficient, in contrast, for instance, with the use of the Mandel parameter or the
P-distribution. Inthecaseofentanglement,wewillmakeuseofthenegativity[34],whichisa
sufficientcriterionitself,andofalowerboundfortheentanglementofformation[35]. Theanal-
ysiswillbeperformedbyconsideringthedimer-excitonsystembothinthepresenceandinthe
absenceofdecoherenceeffectsinducedbylow-energymodesinthephononenvironment. Itwill
bealsointerestingtomonitorthequantumcharacterofthebathbyquantifyingtheentanglement
betweenthebathitselfandthesystem.
2
2. Results
2.1. Model
InRef.[24],itwasshownthatinprototypedimerspresentinavarietyofbiologicalsystems,
efficientvibration-assistedenergytransferinthesub-picosecondtimescaleandatroomtemper-
aturecanappear. Itwasalsoshownthatnon-classicalfluctuationsofcollectivepigmentmotions
are dynamically created. Based on these observations, it was suggested that a connection may
exist between these fluctuations and the high efficiency of the process. The model employed
consistsofaneffectivedimercoupledtoanundampedbosonicmode. Despiteitssimplicity, it
capturestheessentialfeaturesoftheproblem.
Let us just briefly recall the physical model. A dimer is composed of two chromophores
whoseHamiltonianreads
(cid:88)
H = εσ+σ−+V(σ+σ−+σ+σ−), (1)
el i i i 1 2 2 1
i=1,2
whereε istheenergyoftheexcitedlevelofthei-thchromophoreandVistheinter-chromophore
i
coupling. The operators σ+ (σ−) create (annihilate) an electronic excitation at site i and are
i i
expressedintermsofPaulimatrices,σ±=(σ ±iσ )/2.Notethatσ+σ−andσ+σ−areoccupation
i x y 1 1 2 2
numberoperatorswitheigenvalues0,1,whileNˆ (cid:17)σ+σ−+σ+σ− isaconstantofmotionofH
1 1 2 2 el
with eigenvalues 0, 1, 2. The dimer is strongly coupled to a quantized vibrational mode of
frequencyω ,thephononHamiltonianbeing
vib
H = ω (b†b + b†b ), (2)
vib vib 1 1 2 2
whereb† (b ), j=1,2,arebosonicoperatorswhichcreate(annihilate)onephononofthevibra-
j j
tionalmodeofthei-thchromophore,sothatonecandefinethecorrespondingnumberoperators,
nˆ (cid:17)b†b . Finally,theelectronicexcitedstatesinteractwiththeirlocalvibrationalenvironments
j j j
withstrengthg. ForthecorrespondinginteractionHamiltonianonehas
(cid:88)
H =g σ+σ−(b†+b). (3)
el-vib i i i i
i=1,2
CombiningEqs. (1-3)andrestrictingthedynamicstooneelectronicexcitation,oneobtainsthe
effectiveHamiltonian
ε −ε g
H = 1 2σ +Vσ − √ σ (b† +b )+ω b†b , (4)
ex-vib 2 z x 2 z − − vib − −
√
whereσ =σ+σ−−σ+σ− andσ =σ+σ−+σ+σ− andwhereb† =(b†−b†)/ 2istherelative
z 2 2 1 1 x 1 2 2 1 − 1 2
displacementphononmode. AdetailedderivationofEq. (4)isgivenintheappendix.
2.2. Exciton-vibrationcorrelations
The quantum character of the bosonic field was analyzed by the authors of Ref. [24] using
boththeGlauber-SudarshanP(α)-functionandtheMandelfactorQ=((cid:104)nˆ2(cid:105)−(cid:104)nˆ(cid:105)2)/nˆ −1. Both
P(α)<0andQ<0areusedasasufficientcriteriontoverifythepresenceofquantumnessinthe
system.Forinstance,anegativevalueofP(α)impliesthatthedensitymatrixcannotbeexpressed
as a statistical mixture of coherent states and, then, does not admit a classical interpretation.
3
Taking the numerical parameters from the cryptophyte antennae phycoerythrin (PE545) it was
shownthatbothQ(t)andtheP(α)-functionexhibitpartialnon-classicalbehaviour,thatis,there
are regions of time for Q(t) and regions of α for P where quantum fluctuations are detected.
Theseregionswouldvanishintheabsenceofdipolecoupling(V =0).
A different approach to nonclassicality has emerged in the literature, which focusses on
information-theoretic aspects of correlations and has been shown to be deeply inequivalent to
thequantumphasespaceandquasi-probabilitydistributioncriteria[36]. Thus, theconceptsof
entanglement and quantum discord, considered as fundamental resources in many quantum in-
formationprotocols,offeranindependentandconceptuallystrongeralternativeofverifyingthe
quantumcharacteroftheprocessunderstudy.
Inordertodetecttheentanglementbetweenthedimerandthebosonicmodewewillresort
to the Peres-Horodecki criterion, which states that if the density matrix fails to be positive un-
der partial transposition [32], then we are necessarily in the presence of entanglement. As a
quantifier, we use the negativity of entanglement E , which amounts to the sum of the nega-
N
tiveeigenvaluesofthedynamicaldensitymatrixafterpartialtransposition. Indeed,anybipartite
statehastheform(cid:37)=(cid:80) c |i(cid:105)(cid:104)j|⊗|k(cid:105)(cid:104)l|andthepartialtransposemapI⊗T((cid:37))transformsit
into(cid:37) = (cid:80) c |i(cid:105)(cid:104)j|⊗ij|kll(cid:105)(cid:104)kij|k.lAsanybipartiteseparablestatecanbewritten, bydefinitionof
ijkl ijkl
separability, as (cid:37) = (cid:80)ipi(cid:37)(Ai) ⊗(cid:37)(Bi), I ⊗T(ρ) would map it into (cid:37)TB = (cid:80)ipi(cid:37)(Ai) ⊗((cid:37)(Bi))T, which
is a perfectly acceptable density matrix. Then, all the eigenvalues λi of (cid:37)TB are real, positive,
andobey(cid:80)iλi = 1. If, ontheotherhand, someoftheeigenvaluesof(cid:37)TB arenegative, wecan
concludethat(cid:37)doesnotadmitafactorizedform. Basedontheseconsiderations,thequantitywe
aregoingtocalculateisanentanglementwitness,eventhoughitisnotapropermeasure,aside
fromsomespecialcases(arecentexamplecanbefoundinRef. [37]). Statesthatareentangled
even though their negativity vanishes are known as bound entangled states, their main charac-
teristicbeingthatitisnotpossibletoobtainpureentangledstatesfromthembymeansoflocal
operationsandclassicalcommunication.
Thenegativityisalsousefultocalculatealowerboundfortheentanglementofformationof
a bipartite state [35]. Indeed, entanglement of formation represents one of the most meaning-
ful measures of entanglement, as its regularized version quantifies the minimal cost needed to
preparequantumstatesintermsofEPRpairs.
Together with entanglement, it is also of interest to monitor the behaviour of quantum dis-
cordD,whichisamoregeneraldefinitionofquantumnesswithrespecttoentanglement,being
nonzeroevenincaseoffactorizedstateswhosecorrelationsdonotadmitanyclassicalinterpre-
tation [33]. The definition of discord is given in the appendix. Here we anticipate that, given
twopartiesAand Bthequantificationofthecorrelationsbetweenthemisthegoalofourstudy,
itmeasurestheminimumamountofdisturbanceintroducedinthestateofparty Abecauseofa
measurementprocessperformedonpartyB.
ThebehaviourofthesetwoquantitiesisshowninFig. 1asafunctionoftime. Thesystem
is prepared at t = 0 in the Gibbs (thermal) state at room temperature T = 270 K of one of the
vibrationalmodes,whichisinitiallyuncorrelatedwiththedimerstate|X+(cid:105):
ρ(0)=|X+(cid:105)(cid:104)X+|⊗(cid:37)th. (5)
vib
(cid:112)
Here, H |X (cid:105) = λ |X (cid:105), where λ = ± (ε −ε )2+4V2/2. We assume that the frequency of
el ± ± ± ± 1 2
the mode is much larger than the thermal energy scale ω (cid:29) K T. The time evolution is
vib B
calculatedbysolvingtheLiouville-vonNeumannequationρ˙ = −i[H ,ρ]. Inprinciple,as
ex-vib
wedealwithaninfinite-dimensionalsystemtheeigenstatesofwhicharenotGaussianfunctions,
4
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
t(ps)
Figure1: Quantumdiscord(red, dashed), entanglementnegativity(blue, dotdashed)andentanglementofformation
lowerbound(black)asafunctionoftime.Theparametersused,takenfromRefs.[26,27]arethefollowing:ε1−ε2=
1042cm−1, V=92cm−1, ωvib=1111cm−1,andg=267.1cm−1.
we would need to calculate an infinite number of matrix elements in order to determine the
exact full dynamics of the state and its correlations. However, taking into account that, in the
regime ω (cid:29) K T, at the initial time only a few matrix elements are significantly populated,
vib B
a truncation in the number of excitons is a very good approximation. The truncation scheme
consistsofneglecting allthematrixelementsbetweenthe thresholdn˜ and n˜ +1. Inthemodel
understudywecansafelytaken˜ =5,astheresultsdonotchange(withinthemachineerror)for
higherthresholds.
Negativity of entanglement is calculated by applying transposition to the dimer part of the
densitymatrixwhileleavingthebosonicmodeunchanged. Inthecaseofquantumdiscord,the
dimer represents the part under measurement. In principle, an optimization over a complete
positiveoperatorvaluedmeasure(POVM)withelements{EB}shouldbeperformedinorderto
j
gettheoptimalmeasurement. However,forthesakeofsimplicity,wewilllimitourselvestothe
classoforthogonalprojectors. Inallthecasesknowninliterature,orthogonalprojectorsgivea
verytightboundandcanbesafelyusedwithoutanyappreciablequalitativechange.
Working in the ω (cid:29) K T regime implies that the dynamics is largely dominated by the
vib B
coherentoscillationbetween|X+,0(cid:105)and|X−,1(cid:105),whichabsorbsmostofthespectralweightofthe
whole density matrix. We then expect, out of the exact dynamics, a kind of “two-qubit” Rabi-
likeoscillationthatunavoidablygeneratesentanglement.Asitcanbeobserved,thequantumness
witnessedbythepresenceofnegativityandquantifiedbybipartitediscordandentanglementof
formation,isactuallypresentforanyt>0,eveninthetimewindowswhentheQ-parameterand
the P-distribution fail to detect it, that is, when the Rabi-like oscillations reach their minimum
[24]. Aroundthoseregions,thepersistenceofquantumeffectsclearlyindicatesthat,beyondthe
mainoscillation,themulti-modecharacterofthevibrationplaysanimportantrole.Interestingly,
Fig. 1 shows that all quantifiers are close to the values corresponding to maximally entangled
statesformostofthetimeintervalanalyzed(cf. thevalue1/2forthenegativity). Wealsoem-
phasizethat,sinceentanglementissuccessfullydetectedbythenegativityforanyt,theproblem
ofdeterminingthepossibleoccurrenceofboundentanglementisremoved[32].
5
2.3. Single-modedescriptionofthermalnoise
So far, we have discussed the interaction of the dimer with a single high-energy vibronic
mode. Ingeneral, low-frequencyphononicmodes arealsopresentand cannotbeneglected, as
theirexistence,albeittakenintoaccountatahigherperturbationorder,wouldcausedecoherence
anddissipationofthesystemunderinvestigation.
A phonon bath is described by a collection of independent harmonic oscillators that can
cause incoherent transitions between the system eigenstates. The position of each phonon is
indeedcoupledtotheexcitonoperatorσ ,whileitisdecoupledfromthevibrationalmode. The
z
bathHamiltoniancanbewrittenas
(cid:88)
H = ω b†b , (6)
B k k k
k
whilethesystem-bathinteractionHamiltoniantakestheform
(cid:88)
H = g (σ ⊗11 )(b†+b ). (7)
I k z vib k k
k
In principle, a hierarchical expansion of the interaction could be employed to obtain the
reduced system dynamics [38]. The second order (Bloch-Redfield) perturbation theory would
greatly simplify the calculation. However, due to the strong detuning between the system and
the bath, non-Markovian effects, which would not be captured by that method, are expected
to be relevant. Remarkably, a qualitative description of the phenomenon can be obtained by
drastically simplifying the approach. In the absence of the environment, as illustrated in the
previoussection,thedynamicsisdeeplyinfluencedbytheapproximatedegeneracyofthelevels
|X+,0(cid:105)and|X−,1(cid:105). Astheinitialpopulationof|X+,0(cid:105)atroomtemperatureiscloseto1,coherent
oscillations between those two levels are observed which, among other effects, also determine
the establishment of the quantum correlations described in Fig. 1. As the eigenmodes of the
bath lie in a region of the energy spectrum that is far apart from the frequencies of the closed
system,itisnaturaltoaskwhetherandtowhatextenttheinternalstructureoftheenvironment
matters. To this end, we employ a minimal approach replacing the whole environment with a
single,low-energybosonicmodek . Since,becauseoftheapproximationmethodused,weare
0
notinthepresenceofatruebath,asinanyfew-bodyproblem,acontinuousflowofinformation
(whichisthecauseofoscillations)betweenthesystemandthelow-frequencymodeisexpected
totakeplaceinsteadoftherelaxationbehaviourtypicalofdecoherence. Noticethatinthisway
thenon-Markoviancharacteroftheevolution(evenifinanapproximateform)iskept.
Usingthecouplingg andthefrequencyω ofk asfreeparameters, weexploreddifferent
0 0 0
regimesandfoundbehavioursthatareinqualitativeagreementwiththewholebathcase. InFig.
2 we plot the population of the excitonic eigenstate |X (cid:105), i.e. P = (cid:80) p (t) as a function
− X− n n,X−
oftimeandofg forω = 10−2ω . Intherangeofg chosen(centeredapproximatelyaround
0 0 vib 0
g =10−1g),weobservethetransitionfromthecoherentregimetotheincoherentone,wherethe
0
spectralweightofthe|X+,0(cid:105)→|X−,1(cid:105)isreduced. Asalreadypointedout,withinoursimplified
model, P does not reach a true stationary state. However, there is the clear tendency to get
X−
stabilized around a plateau. As expected, if the frequency of that mode is too close to ω ,
vib
where the weak-coupling approximation breaks down, the single mode is not able to capture
the essential features of the whole bath. This is illustrated in Fig. 3, where we have chosen
ω =10−1ω . Inthisregime,decoherenceisexpectedtotakeplacebeforecoherentoscillations
0 vib
areestablished[24],whileweobservehigh-visibilityoscillations.
6
Figure 2: Top panel: PX− as a function of time and g0. The system parameters are the ones given in Fig. 1, and
ω0=10−2ωvib.Lowerpanel:PX−asafunctionoftimeforg0=0(red)andforg0=10−1g(black).
Figure3: PX−asafunctionoftimeandg0forω0=10−1ωvib.
7
Figure4:NegativityofentanglementENasafunctionoftimeandg0forω0=10−2ωvib.
Onceestablishedtheconditionsunderwhichmodelingthebathasasinglemoderepresentsa
suitableapproximation,weusethistechniquetostudythebehaviourofentanglementnegativity,
represented in Fig. 4, for the case where decoherence is expected not to completely suppress
thecoherentoscillation(Fig. 1). Forthesakeofclarity,here,wewillomitthecomplementary
discussionaboutentanglementofformationanddiscord,astherearenoqualitativedifferences.
As in the unperturbed case, we observe an initial growth after which the indicator starts going
down. The important point is that E remains positive at any time, showing the resilience of
N
entanglement against noise. It may also be interesting to see whether the decrease of dimer-
vibrationentanglementissomewhatcompensatedbythecreationofentanglementbetweenthe
dimerandthebath.Actually,apartfromsomespecialvaluesofg andω andpossiblybecauseof
0 0
theroughnessofthesingle-modeapproximationperformed,thisdoesnothappen(seeFig.5),or,
atleast,entanglementnegativitydoesnotrevealit. Thismeansthatagloballossofquantumness
takesplaceeveninthesingle-modeapproximation. Letusstressthat,evenifwehavemodelled
thebathasagenuinequantumsystem,theresultsindicatethatitactuallybehavesasaclassical
sourceofnoise.Infact,itinducesdecoherenceintheexciton-vibrationpartofthemodelwithout
gettingquantumcorrelateditself. Noticethatwehaveconsideredaverysmallenvironmentthat
unavoidably presents recurrences regimes and back-flow of information. The absence of bath-
system quantum correlations is an interesting result, even if not completely unexpected due to
thefactthatthedistributionofentanglementamongmanypartiesisconstrainedbytheproperty
ofmonogamyandthemajorquantityofentanglementisalreadyestablishedbetweenthedimer
andtheresonantmode[39].
3. Conclusions
Understandingtheveryfundamentalmechanismresponsibleforhigh-efficiencyenergytrans-
ferinphotosynthesisisexpectedtoleadtobothfundamentalandpracticalimplications. Onthe
one hand, excitation energy distribution is unavoidably influenced by the presence of environ-
mentaldegreesoffreedom,while,ontheotherhand,theroleplayedbyquantummechanicsin
biologicalstructureshasyettobefullyunderstood.
Wehaveinvestigatedthepresenceofquantumcorrelationsinadimer-excitonsystemusing
negativity of entanglement and quantum discord. From a qualitative point of view, we have
foundtracesofquantumnessatanytimeofinterest. Thisimpliesthatquantumeffectsareeven
deeperthanwhatindicatedbythetimebehavioroftheMandelparameteroroftheGlauberquasi-
probabilitydistribution. Ourresultssuggestthattheinformationconveyedbyentanglementand
discordisricherandcouldbeextendedtodifferentmodelsandworkingregimes. Itseemsthat
8
Figure5: Negativityofentanglementbetweenthedimerandthelow-frequencymodeasafunctionoftimeandg0for
ω0=10−2ωvib.
thephase-spacecharacterizationofnon-classicalityismoresensitivetothemaincontributionto
thedynamics,whileentanglementandquantumdiscordalsocapturethemulti-modestructureof
thebosonicmode. Letusstressthatthephase-spaceparameters,aswellasentanglementnega-
tivity(apartfromsomespecialcase[37])aresufficientcriteriafortheexistenceofquantumness,
while discord is both sufficient and necessary. Then, quantum correlations are established im-
mediatelyaftertheinteractiontakesplace. Letuspointoutthatwehaveobservedacorrelation
betweentransportandnon-classicalproperties. Thisdoesnotnecessarilyimplytheexistenceof
a causal relation, which we would not able to prove, and indeed such kind of answer has not
beenfoundyetinalltheexistingliteratureofthefield. Nevertheless, ourresultspointoutthat
quantumcorrelationscanplayasignificantroleandmaystimulatefurtherstudiesonthesubject,
eventuallyaddressedtounderstandwhetherafunctionalroleexistsornot.
It is also important to set the limits under which non-classicality is robust against thermal
noise. Weproposedasimplifiedapproachtothisproblembasedontheuseofasinglemodeto
mimic the role of the environment. The computational advantage of this method, compared to
hierarchicalexpansions,isevident. Asexpected,thissimplifiedapproachisonlymeaningfulin
thetheweakcouplinglimit.Wetestedthemethodinoursystem,foundtheregimewhereitcanbe
applied,andusedittoassessthequantumcharacterofthedynamicsalsointhepresenceofnoise.
Unexpectedly, despite the high-degree of non-Markovianity and and back-flow of information
inducedbysuchasmallenvironment,negligiblequantumcorrelationsaredetectedbetweenthe
systemandthebathitself. Wehavefoundaregimewhereevenasinglemodeactsasaclassical
source of noise as distribution of entanglement among many parties is subject to monogamy
restraints.
Finally, letusstressthattheverysamemodelstudycanalsobefoundindifferentphysical
contexts. For instance, it could describe electron-phonon interaction in metals. It might also
beinterestingtoanalyzethedynamicalbehaviourbyconsideringdifferentsystemparametersin
ordertoestablishwhethertheobservedphenomenonisageneralcharacteristicofthemodelor
itrequiresspecificexperimentalconditiontobematched.
9
Acknowledgements
WeareindebtedtoMarioRasettiforusefuldiscussionsandcomments.Thefinancialsupport
of Compagnia di San Paolo (Torino, Italy) in the frame of the INRIM project on “Quantum
Correlations”isgratefullyacknowledged.
AppendixA
ThemodelintroducedthroughEqs. (1-3)isdefinedinthetensorproductHilbertspaceH=
H ⊗H ⊗F ⊗F ,whereH ,H arethetwo-dimensionalHilbertspacesofthechromophores
1 2 1 2 1 2
andF ,F arethe∞-dimensionalFockspacesofthephonons. Theanalysissimplifiesconsider-
1 2
abl(cid:112)ybyintroducingtheeigenstates|X±(cid:105)ofHel, Hel|X±(cid:105)=λ±|X±(cid:105),withe√nergysplittingλ+−λ−
= (ε −ε )2+4V2,andthecollectivephononmodesb† = (b†±b†)/ 2,withb† (b†)corre-
1 2 ± 1 2 + −
spondingtothecenter-of-mass(relativedisplacement)phononmode.Inviewofthepropertiesof
Nˆ,theeffectivechromophoresHilbertspacereducestothesingletwo-dimensionalone-particle
(orspin 1)spaceH,withH ⊗H (cid:55)−→H. Furthermore,sincethecenter-of-massmodeb† isnot
2 1 2 +
coupledtotheelectronicdegreesoffreedom,onlytherelativedisplacementbosonicoperatorb†
−
isrelevanttothesystemdynamics,sothatthephononspacemapsintothesingle∞-dimensional
FockspaceF,i.e.,F ⊗F (cid:55)−→F. Eqs. (1-3)arecombinedresortingtotheabovesimplifications
1 2
and hence giving the effective exciton-vibration Hamiltonian introduced in Eq. (4). There, the
Paulimatricesσ andσ liveinH.
z x
AppendixB
Given a bipartite density matrix (cid:37) and its reduced states (cid:37) = Tr {(cid:37)} and (cid:37) = Tr {(cid:37)},
A B B A
discordcanbedefinedasthemeasureofhowmuchdisturbanceisintroducedwhentryingtoget
informationaboutpartyAwhenpartyBismeasured[33]. Itisdefinedas
D =I((cid:37))−J , (.1)
A:B A:B
whereI((cid:37))=S((cid:37) )−S((cid:37) )−S((cid:37))isthequantummutualinformation,obtainedfromitsclassical
A B
counterpartreplacingtheShannonentropywiththevonNeumannentropyS((cid:37)) = −Tr{(cid:37)log(cid:37)},
andwheretheclassicalcorrelationsaregivenby
J =max[S((cid:37) )−S(A|{EB})], (.2)
A:B A j
{EB}
j
with the conditional entropy S(A|{EB}) = (cid:80) p S((cid:37) ), p = Tr (EB(cid:37)) and where (cid:37) =
j j j A|EBj j AB j A|EBj
EB(cid:37)/p isthedensitymatrixafteraPOVMwithelements{EB}hasbeenperformedonparty B.
j j j
Noticethatthedefinitionofdiscordisnotsymmetricundertheexchangeofthetwoparties. It
is even possible to find states that are quantum-classical, that is, states that behave as quantum
objectsifoneofthepartiesisobservedandasclassicalobjectsbyobservingtheotherparty.
References
References
[1] G.S.Engeletal.,Nature(London)446(2007)782.
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