Non-classicality of molecular vibrations activating electronic dynamics at room temperature

1.6 MB·English

by  Edward J. O'Reilly

Non-classicality of molecular vibrations activating electronic dynamics at
room temperature
Edward J. O’Reilly1 and Alexandra Olaya-Castro1,a)
Department of Physics and Astronomy, University College London, Gower Street, London,
WC1E 6BT United Kingdom.
Advancingthedebateonthepossibleadvantagesofquantumeffectsinbiomolecularsystemsrequiresillustra-
tions of biomolecular prototypes that can exhibit and use non-classical phenomena to aid performance. Here
weinvestigatenon-classicalfluctuationsinmoleculardimerspresentinphotosyntheticlight-harvestinganten-
nae. We show that exciton-vibrational resonances enable non-classical occupation fluctuations of the driven
collective high-energy vibration at room temperature. This phenomenon is a manifestation of strong mod-
ulation of low-phonon occupation levels through the transient formation of hybrid exciton-vibration states.
For the prototype considered, direct quantitative relations seem to hold between non-classicality and optimal
exciton population transfer.
3
1
0
The experimental investigation of quantum effects in and are of particular interest as they represent sys-
2
photosynthetic energy transfer1–5 has inspired exciting tems where, although counter-intuitive, exciton coher-

n research towards asserting the relevance of non-classical ence beating has been probed23. Insights into the im-
a
behaviourinthebiologicalfunctionofthemolecularcom- portanceofresonancesbetweenexcitonenergymismatch
J
ponents of living organisms. Theoretical works have and vibrational frequencies for efficient energy trans-
9
made emphasis on the quantumness exhibited by elec- fer can be gained from F¨orster theory24. However, the
2
tronic degrees of freedom6 and on the key role of envi- widerimplicationsofsuchresonancesforoptimalspatio-

] ronmentalfluctuations7,8. Inordertounderstandthefull temporal distribution of energy16,25, for manipulation of
h range of possible non-classical phenomena present dur- excitoncoherence13,14,16 andfornon-adiabaticexcitation
p ing room-temperature dynamics of electronic excitations dynamics18 have only started to be clarified. An is-
-
t in light-harvesting antennae it is important to expand sue that remains to be tackled, is the extent to which
n
the analysis to the molecular motions that play a fun- such resonances can expose the non-classical features
a
u damental role in driving exciton transport9. Research of the molecular motions activating transport. In the
q in this direction is timely given that experiments10,11 dimer investigated here, commensurate energies of ex-
[ and theoretical works12–18 are highlighting the impor- citon gap and discrete vibrations imply that after in-

tance of concerted electronic and vibrational dynam- put of exciton population, effective energy distribution
1
ics in energy transfer. Moreover, recent experimen- proceeds by driving the collective relative displacement
v
0 tal techniques able to manipulate vibrational states19,20 mode out of thermal equilibrium towards a non-classical
7 and probe their quantum properties may indeed pro- state. Non-classicalityofthestateofthisrelativemotion
9 vide experimental access to the non-classicality of vi- isunambiguouslyindicatedbysub-Poissonianoccupation
6 brational modes in biomolecular systems. Here we con- statistics26 or alternatively by strong modulation of low-
1. sider a prototype molecular dimer with an structured phononoccupations27 andisamanifestationoftransient
0 exciton-phonon interaction and investigate non-classical collectivecoherentbehaviourbetweenexcitonsandvibra-
3 behaviour of molecular vibrations that steer energy dy- tions. Low-energy thermal fluctuations stabilize energy
1
namics and exciton coherences in this molecular unit. distribution to the lowest exciton state before destroy-
:
v Whilepreviousstudieshaveconsideredcoherentexciton- ing non-classicality of the collective mode assisting the
i vibration dynamics in molecular dimers14,17,21, it is un- process. Meanwhile, coherent interaction between elec-
X
clearhowthisdeviatesfromclassical(wave-like)coherent tronic degrees of freedom and such a mode manifests it-
r
behaviour and if so whether there are quantitative re- selfinthebeatingofexcitoniccoherenceswithfrequency
a
lationships between non-classicality and effective energy components that vary depending on the coupling to the
transfer. thermalbath. Whentime-averagingoverarelevanttime
Theprototypeofinterestconsistsoftwochromophores scale, we find direct quantitative relations between pop-
coupled by weak electronic interactions and subject to ulation transfer and the degree of non-classicality as in-
the influence of both a low-energy thermal background dicated by the average of negative values of the Man-
and a discrete high-frequency intramolecular vibration. del Q−parameter26. This relationship holds even when
The key aspect of this molecular vibration is that its theelectronicsystemisinitializedinastatisticalmixture
energy matches closely the exciton gap but it is much of exciton states. Hence, efficient photosynthetic energy
largerthanthethermalenergyscale. Dimersofthistype transport could take advantage of non-classicality.
naturally occur in several light-harvesting proteins12,22
The field of quantum optics has developed a solid
framework to quantify the non-classicality of bosonic
fields28. It therefore provides useful conceptual and
a)Electronicmail: [email protected] quantitative tools to investigate non-classicality of the
2
√
harmonic vibrational degrees of freedom of interest in with the bosonic operators b(†) = (b(†) −b(†))/ 2, cou-
1 2
thiswork. Fromtheperspectiveofquantumoptics, non- ples to the excitonic system. The centre of mass mode
classicality arises if the state of the system of interest decouplesfromtheelectronicdegreesoffreedom. Theoc-
cannot be expressed as a statistical mixture of coher- cupation of the relative displacement mode of frequency
ent states defining a valid probability measure, this then ω is denoted by n and the non-classical properties of
vib
leads to non-positive or highly singular values of the such mode is the subject of interest in this article.
Glauber-SudarshanP-functionortheWignerfunction29.
Ourprototypesystemcorrespondstothecentraldimer
However, experimental characterization of these distri-
of the phycoerythrin 545 (PE545) antennae found in
butions is not always a straightforward task. Alterna- cryptophyte marine algae30. Exciton coherence beating
tively, it is of interest to investigate signatures of non- involving this dimer has been probed4 and a structured
classicality. For a single-mode field, a sub-Poissonian oc-
spectraldensityhasbeenphenomenologicallydetermined
cupationnumberdistributionquantifiedbytheMandel’s via linear spectroscopy31. The electronic parameters for
Q−parameter26 isasignatureofdeviationsfromclassical the dimer taken from reference31 are ε = 19574cm−1,
1
behaviour. The Mandel Q−parameter characterizes the ε = 18532cm−1 and V = 92.2cm−1. The intramolecu-
2
departure of the occupation number distribution P(n)
lar mode that is separated from the set of vibrational
from Poissonian statistics through the inequality degrees of freedom has frequency ω = 1111cm−1
vib
(cid:104)nˆ2(cid:105)−(cid:104)nˆ(cid:105)2 being quasi-resonant with the exciton energy splitting
Q= −1<0 (1) ∆E = 1058.2cm−1. The coupling strength of each
(cid:104)nˆ(cid:105) √
chromophore to the this mode is g = ω 0.0578 =
vib
where (cid:104)nˆ(cid:105) and (cid:104)nˆ2(cid:105) denote the first and second moments 267.1cm−1 obtained from30,31, and the thermal energy
of the bosonic number operator nˆ respectively. Van- scale at room temperature is kBT ≈ 200cm−1. We con-
ishing Q indicates Poissonian number statistics where sider the influence of the low energy thermal fluctua-
the mean of nˆ equals its variance (such is the case of tions via a thermal bath HB described by a continuous
a coherent state of light). For a chaotic thermal state distribution of harmonic modes, each linearly and inde-
one finds that Q = (cid:104)nˆ(cid:105) > 0 indicating that particles pendently coupled to the excited electronic states Hel
are ‘bunched’. The departure from classical (wave-like) withcouplingstrengthcharacterisedbyaDrudespectral
behaviour is characterised by Q < 0 or particle num- density J(ω) = (λΩcω/π)/(ω2+Ωc2)2 with reorganiza-
ber distributions where the variance is less than the tion energy λ = 110cm−1 and cut-off frequency Ωc =
mean, as is the case for Fock states. The occupation 100cm−132, hence the stren√gth of the system-bath inter-
fluctuations associated with a sub-Poissonian distribu- action is parametrized by λΩc33. Since J(ωvib) (cid:28) g2,
tion originate from the quantized nature of the field and this does not over-account for the influence of the res-
cannot be described in terms of an underlying classi- onant mode on the electronic system. Importantly, al-
cal stochastic process. A criteria that directly connects though the electronic coupling V is the smallest energy
non-classical occupation fluctuations and the negativity scale in the prototype dimer, the exciton dynamics and
of a quasiprobability distribution is an inequality indi- phonon fluctuations here investigated will not be prop-
cating modulation of nearest number occupations27. In erly described by F¨orster theory. In order to accurately
this work we use the above framework to quantify non- account for the effects of the thermal bath we have ex-
classicality of the undamped vibrational modes support- tended the hierarchical expansion of the dynamics34–36
ing exciton transport in the molecular dimer of inter- to explicitly include the quantum interactions between
est. We consider each chromophore with an excited elec- electronic excitations and the selected mode. This ap-
tronic state of energy ε strongly coupled to a quantized proach allows us to account accurately for the effects of
i
vibrational mode of high-frequency (ω > K T) de- thelow-energythermalbath. Incalculatingdynamicswe
vib B
scribed by the bare Hamiltonians H = (cid:80) ε σ+σ− consider an initial state, in which the upper energy exci-
el i=1,2 i i i
ton eigenstate is rapidly excited while both high-energy
and H = ω (b†b + b†b ) respectively. Inter-
vib vib 1 1 2 2 quantized vibrations and low-energy thermal modes are
chromophore coupling is generated by dipole-dipole in-
initially in thermal equilibrium at room temperature.
teractions of the form H = V(σ+σ− +σ+σ−). The
d-d 1 2 2 1 Truncating the quantized modes at an occupation Fock
electronic excited states interact with their local vi-
levelM =6adequatelydescribesbothreduceddynamics
brational environments with strength g, linearly dis-
of the collective quantized mode and the electronic dy-
placing the corresponding mode coordinate, H =
el-vib namics. Adescriptionofthenumericalmethodologycan
g(cid:80) σ+σ−(b†+b ). In the above b†(b ) creates (an-
i=1,2 i i i i i i be found in the appendix.
nihilates) a phonon of the vibrational mode of chro-
Hamiltonian dynamics and non-classicality.- We first
mophore i while σ± creates or destroys an electronic
i illustrate the quantum coherent dynamics of excita-
excitation at site i. The eigenstates |X(cid:105) and |Y(cid:105) of
tion when coupled to the undamped quasiresonant high-
H + H denote exciton states with energy splitting
el d-d (cid:112) energyvibrationalmodesintheabsenceofthelow-energy
given by ∆E = (ε −ε )2+4V2. Transformation into
1 2 thermal bath. This allows us to identify relevant time
collective mode coordinates shows that only the mode
scales in the evolution of a state with no initial exci-
corresponding to the relative displacement of the modes
tonic superpositions: ρ(t ) = |X(cid:105)(cid:104)X|⊗(cid:37)th which writ-
0 vib
3
ten in the basis of exciton-vibration states of the form
(cid:80)
a |X,n(cid:105) becomes ρ(t ) = P (n)|X,n(cid:105)(cid:104)X,n|. Here
0 n th
n denotes the phonon occupation number of the collec-
tive mode while P (n) denotes the thermal occupation
th
ofsuchlevel. Weinvestigatethepopulationofthelowest
(cid:80)
excitonic state ρ (t) = (cid:104)Y,n|ρ(t)|Y,n(cid:105), the abso-
YY n
lute value of the coherence ρ (t) = (cid:104)X,0|ρ(t)|Y,0(cid:105)
X0−Y0
whichdenotestheinter-excitoncoherenceintheground-
state of the collective vibrational mode, and the non-
classicality given by negative values of Q(t). Hamilto-
nianevolutiongeneratescoherenttransitionsfromstates
|X,n(cid:105) to |Y,n + 1(cid:105) (see Figure 1(a)) with a rate f
that depends on the exciton delocalization (|V|/∆(cid:15)), the
0.7
b coupling to the mode g, and the phonon occupation n
0.6
(cid:112)
i.e. f (cid:39) g(2|V|/∆ε) (n+1)/2. Since ω (cid:29) K T
0.5 vib B
0.4 the ground state of the effective mode is largely popu-
0.3 lated, such that the Hamiltonian evolution of the initial
0.2 state is largely dominated by the evolution of the state
0.1 |X,0(cid:105). This implies that the energetically close exciton-
0.0 vibration state |Y,1(cid:105) becomes coherently populated at a
0.0 0.2 0.4 0.6 0.8 1.0 rate f (cid:39) g(2|V|/∆(cid:15)), leading to the oscillatory pattern
observed in the probability of occupation ρ (t) as il-
t(ps) YY
lustrated in Figure 1 (b). The low-frequency oscillations
of the dynamics of ρ (t) cannot be assigned to the ex-
YY
0.6 c citon or the vibrational degrees of freedom alone as ex-
0.4
pected from quantum-coherent evolution of the exciton-
0.2
Q(t) 0.0 pculupsa-teiffoenctisivreesmtroidcteesdystoteamt.mFoosrtinns=tan1c,et,hieftpheerimododoef tohce-
-0.2
amplitudeofρ (t)isgivenapproximatelybytheinverse
-0.4 YY
(cid:112)
of(1/2)( (∆E−α)2+(2g24V2/α∆E)−(∆E+α))with
-0.6
α2 =2g2+ω2 and(2g24V2/α2∆E2)(cid:28)1. Coherentex-
0.0 0.2 0.4 0.6 0.8 1.0 vib
citon population transfer is then accompanied by coher-
t(ps) ent dynamics of the inter-exciton coherence |ρ (t)|
X0−Y0
with the main amplitude modulated by the same low-
0.015 frequency oscillations of ρ (t) and a superimposed fast
d YY
oscillatory component of frequency close to ω as can
vib
0.010 be seen in Figure 1 (b). This fast driving component
Q(t) arises from local oscillatory displacements: when V (cid:39) 0
0.005 the time evolution of each local mode is determined by
the displacement operator with amplitude α(t)=2g(1−
exp(−iω t))/ω 37. As the state |Y,1(cid:105)(cid:104)Y,1| is coher-
0.000 vib vib
0.0 0.1 0.2 ently populated, the collective quantized mode is driven
out of equilibrium towards a non-classical state in which
t(ps)
selective occupation of the first vibrational level takes
place. This manifests itself in sub-Poissonian phonon
FIG.1. a,Representationofthepigmentsandproteinenvi- statistics as indicated by negative values of Q(t) shown
ronment of the PE545 complex (Protein Data Bank ID code in Figure 1(c). Similar phenomena have been described
1XG030). Thecentraldimerishighlighted. Alsoshownarethe in the context of electron transport in a nanoelectrome-
energy levels of the exciton-vibration states used to describe chanical system38,39. Importantly, such sub-Poissonian
energy transfer |X,n(cid:105). b, Coherent dynamics of the exciton-
statisticsarisesonlywhentheelectronicinteractionisfi-
vibrationdimer. Oscillatorypopulationofthelowestexciton
nite. For comparison, Figure 1(d) shows that if V = 0,
population ρ (t) (thick blue curve) is accompanied by os-
YY a local excitation drives the mode towards a classical
cillationsoftheinter-excitoncoherenceintheground-stateof
thermal displaced state with super-Poissonian statistics
thevibrationalmode|ρ (t)|(thinredcurve). c,Mandel
X0−Y0
Q-parameterofthevibrationalmodeforbiologicalelectronic (Q(t) > 0). In short, non-classicality of the relative dis-
coupling and d for zero electronic coupling. placement mode quasiresonant with the excitonic tran-
sition arises through the formation of exciton-vibration
states.
Energy and coherence dynamics - Interplay between
4
vibration-activated dynamics and thermal fluctuations Non-classicality - Interaction with a large thermal en-
leads to two distinct regimes of energy transport. For vironment leads to the emergence of classicality. In-
our consideration of weak electronic coupling, the co- deed, as the coupling of the electronic degrees of free-
herent transport regime is determined approximately by dom to the thermal background is increased, the maxi-
√
λΩ ≤2g|V|/∆(cid:15). Here population of the low-lying ex- mum non-classicality of the collective mode as indicated
c
citon state is dominated by coherent transitions between by the most negative value of Q(t) is reduced. How-
exciton-vibration states and the rapid short-time growth ever, the time interval for which the mode remains in
of ρ (t) in this regime can be traced back to coherent a non-classical state is not a monotonic function of λ.
YY
transferofpopulationfrom|X,0(cid:105)(cid:104)X,0|to|Y,1(cid:105)(cid:104)Y,1|. At Formoderatevaluesofλ,therelativedisplacementmode
longertimescalesthermalfluctuationsinduceincoherent spends longer periods in states with non-classical fluctu-
transitions from |X,0(cid:105)(cid:104)X,0| to |Y,0(cid:105)(cid:104)Y,0| with a rate ations i.e. periods for which Q(t)<0 as seen in Figures
√
proportional to λΩ , thereby stabilizing population of 3(a)-(b). Hence relatively medium coupling to a ther-
c
|Y,1(cid:105)(cid:104)Y,1| to a particular value (see Figure 2(a)). In malbackgroundassistsselectivepopulationofstate|Y,1(cid:105)
√
contrast, for λΩ > 2g|V|/∆(cid:15) population transfer to thereby stabilizing non-classicality at a particular level.
c
ρ (t) is incoherent although the dynamics is still ac- An important feature of the results presented in Figures
YY
companied by generation and coherent evolution of ex- 3(a)-3(c)isthereforethatnon-classicalfeaturessurviving
citonic coherences (see Figures 2(g) and 2(h)). In this the picosecond time scale are present across the whole
incoherent transport regime ρ (t) has a slow but con- range of thermal bath couplings. This is consistent with
YY
tinuousrisereflectingthefactthattheincoherenttransi- a slow decay of the exciton-vibration coherence ρ
X0,Y1
tions from |X,0(cid:105)(cid:104)X,0| to |Y,0(cid:105)(cid:104)Y,0| now contribute sig- (notshown). Interestingly,thenon-classicalpropertiesof
nificantly to the transport. However, transfer to ρ (t) the collective mode resemble non-classicality of thermal
YY
isalwaysmoreefficientwiththequasiresonantmodethan states (completely incoherent states) that are excited by
in the situations where only thermal-bath induced tran- asinglequanta41. Thenon-classicalpropertiespredicted
sitions are considered (see dashed lines in Figures 2(d) by Q(t) also agree with the non-classicality predicted by
and 2(g)). The underlying reason is that the system is a parameter proposed by Klyshko27 as is described in
transiently evolving towards a thermal configuration of appendix B.
exciton-vibration states. Hence, in both regimes excita-
Non-classicalityiswitnessedbyexcitonpopulationdy-
tion transfer to the lowest exciton state is achieved by
namics. In order to illustrate this, we now investigate
transientlysharingenergywiththerelativedisplacement
quantitative relations between non-classicality and en-
mode. The transition from coherent to incoherent ex-
ergy transport by considering relevant integrated aver-
citon population dynamics is also marked by the onset
ages in the time scale of coherent exciton-vibration dy-
of energy dissipation of the exciton-vibration system as
namics denoted by τ. For the parameters chosen, this
shown in figures 2(c),2(f) and 2(i). While population
time scale is about half a picosecond and corresponds
transfer is coherent, energy dissipation into the thermal
to the time in which excitation energy would be dis-
bath is prevented (Fig. 2(c) and 2(f)). tributed away to other chromophores16. The time in-
tegrated averages over τ are defined as: (cid:104)F[ρ(t)](cid:105) =
The beating patterns of the coherence ρ (t) re- τ
X0−Y0 1 (cid:82)τdtF[ρ(t)],whereF[ρ(t)]correspondstotheexciton
veals the structured nature of the exciton-phonon inter- τ 0
population ρ (t) and the non-classicality of the collec-
action and witnesses whether there is coherent exciton- YY
tive quantized mode through periods of sub-Poissonian
vibration evolution. Because of this the frequency com-
statistics Q(t)Θ[−Q(t)] as functions of the coupling to
ponents of such oscillatory dynamics vary depending of
the bath λ. As shown in Figure 4 the average exci-
the coupling to the thermal bath as can be seen in
ton population and non-classicality follow a similar non-
Figures 2(b), (e) and (h) . In the coherent regime
monotonic trend as a function of the coupling to the
ρ (t) follow exciton populations with the main am-
X0−Y0
thermal bath, indicating a direct quantitative relation
plitude modulated by the same relevant energy differ-
between efficient energy transfer in the time scale τ and
ence between exciton-vibration states (see Figures 2(b)
the degree of non-classicality. The appearance of max-
and (e)). In contrast, in the regime of incoherent trans-
imal point in the average non-classicality as a function
port the short-time oscillations of ρ (t) (between
X0−Y0
of the system-bath coupling indicates that the average
t = 0 and t = 0.1ps) arise from electronic correlations
quantum response of the relative displacement mode to
due to bath-induced renormalization of the electronic
Hamiltonian40 and as such, the frequency of these short- theimpulsiveelectronicexcitation,isoptimalforasmall
amount of thermal noise.
time oscillations correspond to transitions of renormal-
ized excitons. As the main amplitude is damped, this The above results indicate that the degree of delocal-
exciton coherence retains the superimposed driving at izationoftheinitialexcitonstateiskeytoenablingnon-
a frequency ω . The dynamical features presented in classical fluctuations of the collective quantized vibra-
vib
Figures 2(g) and 2(h) confirm previous findings based tion. This suggests that statistical mixtures of excitons
on perturbative calculations16 and agrees with the time- with finite purity can still trigger such non-classical re-
scales of the exciton coherence beating reported in the sponse. To illustrate this we now consider mixed ini-
two-dimensional spectroscopy of cryptophyte algae23. tial states of the form ρ(t ) = (cid:37) ⊗(cid:37)th where (cid:37) =
0 ex vib ex
5
0.7 0.7 0.7
a d g
0.6 0.6 0.6
0.5 0.5 0.5
(t)Y 0.4 (t)Y 0.4 (t)Y 0.4
Y 0.3 Y 0.3 Y 0.3
Ρ Ρ Ρ
0.2 0.2 0.2
0.1 0.1 0.1
0.0 0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
t(ps) t(ps) t(ps)
0.10 0.10 0.10
b e h
0.08 0.08 0.08
(t)|0 0.06 (t)|0 0.06 (t)|0 0.06
Y Y Y
X0, 0.04 X0, 0.04 X0, 0.04
Ρ Ρ Ρ
| | |
0.02 0.02 0.02
0.00 0.00 0.00
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
t(ps) t(ps) t(ps)
19610 19610 19700
c f i
) 19605 ) 19605 )
-1m -1m -1m 19600
19600 19600
c c c
y( 19595 y( 19595 y( 19500
g g g
ner 19590 ner 19590 ner 19400
E 19585 E 19585 E
19580 19580 19300
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
t(ps) t(ps) t(ps)
FIG.2. Thedynamicsofρ (t)(toprow),|ρ (t)|(middlerow)andEnergyoftheexcitonvibrationsystem(bottomrow)
YY X0,Y0
for three interaction strengths to the low-energy thermal bath: a-c: λ=6cm−1, d-f λ=20cm−1 and g-i: λ=110cm−1
0.4 a 0.4 b 0.4 c
0.2 0.2 0.2
0.0 0.0 0.0
Q(t) -0.2 Q(t) -0.2 Q(t) -0.2
-0.4 -0.4 -0.4
-0.6 -0.6 -0.6
-0.8 -0.8 -0.8
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
t(ps) t(ps) t(ps)
FIG. 3. a-c Dynamics of the Mandel Q-parameter for λ=6,20,110cm−1.
r|X(cid:105)(cid:104)X|−(1−r)|Y(cid:105)(cid:104)Y| with 1/2 ≤ r ≤ 1. The as- tum fluctuations to efficiently populate a target exciton
sociated linear entropy quantifying the mixedness of the state. These non-classical fluctuations are observed for
initial exciton states is given by S = 2r(r −1). The a variety of initial exciton conditions including statisti-
L
time-averaged non-classicality (Figure 5 (a)) and aver- cal mixtures suggesting that such non-classicality may
age population transfer (Figure 5 (b)) follow similar de- be triggered even under incoherent input of electronic
creasing monotonic trends for mixed states. These find- population. These quantum phenomena could be ex-
ings then suggest direct quantitative relationships be- perimentally accessible by transient coherent ultrafast
tween non-classicality and exciton population transport phonon spectroscopy42 which is sensitive to low-phonon
in the relevant time scale. Outlook.- Our work shows populations of high-energy vibrations43. The framework
thatinmoleculardimersoperatingintheweakelectronic here discussed could also be used to give more insight
coupling regime, exciton-vibrational resonances can trig- into the possible non-classical response of vibrational
ger a non-classical response of a quasiresonant collective modes in other chemical sensors44 which are conjec-
high-energy vibration thereby taking advantage of quan- tured to operate through weak electronic interactions45
6
mechanism for thermodynamic control47. Understand-
0.4
inghowquantumphenomenaaffectthethermodynamics
000...123 (cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:88)(cid:230)(cid:224)QΡ(cid:230)(cid:224)(cid:230)(cid:224)Y<(cid:230)(cid:224)Y0(cid:230)(cid:224)(cid:230)(cid:224)(cid:92)(cid:230)(cid:224)ΤΤ(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224) outpvuafnrmonAiefccynGekpiesnrnrshoggeoensywnf4dro8leace,m4ldoml9gnee.eenvfwmeoaroresrindinkotinsstac.o-uivnWsqasurieboaieintnothstym,aifnaoyonklfetdcGhbueAiroleevamgsidnoSvcalcaeoshncuhoutladlKleagsoreplasrltniorovdfafoindRrqseupsieuaonapnrnkt--
port on numerical calculations in the initial stages of
(cid:88) (cid:92)
0.0
this project. Financial support from the Engineering
0 20 40 60 80 100
and Physical Sciences Research Council (EPSRC UK)
Λ(cm-1) is gratefully acknowledged.
FIG. 4. Time integrated averages of exciton popu- 1GregoryS.Engel,TessaR.Calhoun,ElizabethL.Read,Tae-Kyu
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YY
ship, and Graham R. Fleming. Evidence for wavelike energy
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33Javier Prior, Alex W. Chin, Susana F. Huelga, and Martin B. method used to accurately calculate the dynamics of
Plenio. Efficientsimulationofstrongsystem-environmentinter-
these exciton-vibration systems. The system Hamilto-
8
nian H = H ⊗1 +1 ⊗H +H , is defined forthecaseofaDrudespectraldensityandinarescaled
S el vib el vib el-vib
by formulation36 whichadmitstheirefficientadaptivetime-
step numerical integration. Operators ρ˜ are indexed by
n
Hel =(cid:88)N εiσi+σi−+ (cid:88)N Vij(σi+σj−+σj+σi−) (A1) nth±emisulnti-winidthexenntwryhinchh→asenntri±es1n.ikTahnedrned=uc(cid:80)edikdneink-.
ik jk jk
i=1 i,j≤i sity matrix of the system ρ is given by the operator
S
ρ˜ . The coefficients ν and c are the Matsubara fre-
0 ik ik
quenciesandcoefficientsappearinginthe(truncated)ex-
N
Hvib =(cid:88)ωviibb†ibi (A2) ponential decomposition (cid:80)Kk=0cike−νikt of the bath cor-
relation function applied in Ref35. In the above terms
i=1
k > K are truncated with a Markovian approximation
and scheme35. This results in accurate dynamics for K = 1
at high temperatures (βΩ <1). In the present exciton-
H =g σ+σ−⊗(b†+b ) . (A3) c
el-vib i i i i i vibration case the system-bath coupling operator takes
the form Q =σ+σ−⊗1 .
Here, the operators σ± create/annihilate electronic exci- i i i vib
i
tationsatsitei,whileb† (b )create(annihilate)phonons
i i
associatedwiththevibrationalmodeofsitei. ε denotes
i
the excitation energies and V the strength of interac-
ij
tions between sites. ωi is the frequency of the vibra-
vib √
tional mode at site i and g = S ωi the coupling of
i i vib
the mode to its site. S is the Huang-Rhys factor or
i
mean number of phonons in the ‘polaron cloud’ formed Appendix B: B parameter
0
when the mode is fully displaced. The the high energy
modes considered here, we expand the quantized vibra-
tional modes of the system in the basis of single phonon Modulations of adjacent phonon number occupation
(Fock) states (e.g. b = (cid:80)M(n+1)1/2|n(cid:105)(cid:104)n+1|) trun- as quantified by negative B , guarantees negative re-
i n n
cating at M phonons. This system is then coupled to a gions of a quasi-probability distribution27 as is a poten-
continuum of bosonic modes (the thermal background) tially an experimentally accessible quantity. The non-
H =(cid:80) ω b†b , with an interaction classicality indicated by sub-Poissonian statistics is con-
B k k k k
firmed by negative B = 2P(0)P(2)−P(1). This arises
0
H =(cid:88)(cid:88)g (σ+σ−⊗1 )(b† +b ) , (A4) fromselectivepopulationoftheP(1)leveloftherelative
I k,i i i vib k k displacement vibrational mode.
k i
such that a mode with frequency ω is linearly coupled
k
to the electronic excitation at site i with a strength g
k,i
asspecifiedbyaspectraldensityJ (ω)=(cid:80) |g |2δ(ω− 0.2 0.2
i k k,i a b
ω ). We assume the bath is spatially uncorrelated and 0.1 0.1
k
identical for each site. 0.0 0.0
-0.1 -0.1
An infinite hierarchy of coupled differential
equations34–36 is used to express the dynamics of -0.2 -0.2
-0.3 -0.3
the exciton-vibration system density matrix ρ (t).
S 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
These read as,
t(ps) t(ps)
N K 0.2 0.2
(cid:88)(cid:88) c d
∂ ρ˜ (t)=−i/(cid:126)[H ,ρ˜ (t)]− n v ρ˜ (t) 0.1 0.1
t n S n ik ik n
i=1k=0 0.0 0.0
N (cid:32) ∞ K (cid:33) -0.1 -0.1
−i(cid:88) (cid:88) cik −(cid:88) cik [Q ,[Q ,ρ˜ (t)]] -0.2 -0.2
ν ν i i n
ik ik -0.3 -0.3
i=1 k=1 k=1
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
N K
(cid:88)(cid:88)(cid:112) (cid:104) (cid:105)
−i (n +1)|c | Q , ρ˜ t(ps) t(ps)
ik ik i n+
ik
i=1k=0
FIG. 6. B for a) λ=0cm−1, b) λ=6cm−1, c) λ=20cm−1
N K 0
−i(cid:88)(cid:88)(cid:112)n /|c |(cid:16)c Q ρ˜ −c∗ ρ˜ Q (cid:17) . and d) λ=110cm−1
ik ik ik i n− ik n− i
ik ik
i=1k=0
(A5)