Abstract
Advancing the debate on quantum effects in lightinitiated reactions in biology requires clear identification of nonclassical features that these processes can exhibit and utilize. Here we show that in prototype dimers present in a variety of photosynthetic antennae, efficient vibrationassisted energy transfer in the subpicosecond timescale and at room temperature can manifest and benefit from nonclassical fluctuations of collective pigment motions. Nonclassicality of initially thermalized vibrations is induced via coherent exciton–vibration interactions and is unambiguously indicated by negativities in the phase–space quasiprobability distribution of the effective collective mode coupled to the electronic dynamics. These quantum effects can be prompted upon incoherent input of excitation. Our results therefore suggest that investigation of the nonclassical properties of vibrational motions assisting excitation and charge transport, photoreception and chemical sensing processes could be a touchstone for revealing a role for nontrivial quantum phenomena in biology.
Introduction
The experimental demonstration of oscillatory electronic dynamics in lightharvesting complexes^{1,2,3,4,5} has triggered widespread interest in uncovering quantum phenomena that may have an impact on the function of the molecular components of living organisms. In general, however, oscillatory patterns in dynamics are not sufficient argument to rule out classical descriptions of the same behaviour. Indeed, recent work has discussed how classical coherence models can predict electronic coherence beating^{6,7}. Therefore, an important challenge for the growing field of quantum effects in biomolecules is to clearly identify which quantum features with no classical counterpart may manifest in these systems and how they may influence the process of interest.
The question of nonclassicality of the dynamics of electronic excitations in lightharvesting systems has been addressed by investigating Leggett–Garg inequalities^{8}. This work concludes that, under Markovian evolution, temporal correlations of the observables of individual pigments should violate classical bounds, and in consequence certain classical theories are unsuitable to describe electronic dynamics. Other works have investigated the quantumness of the electronic degrees of freedom^{9,10,11,12,13}. Despite these efforts, it is still far from understood which nonclassical phenomena are directly correlated with efficient energy distribution in a prototype lightharvesting system.
What is clear is that exciton energy transport depends not only on the topology of electronic couplings among pigments but is critically determined by exciton–phonon interactions: molecular motions^{14} and environmental fluctuations^{14,15,16} drive efficient transport processes in lightharvesting antennae. In fact, it is well known that exciton–phonon interactions in these complexes have a rich structure as a function of energy and generally include coupling to both continuous and discrete modes associated to lowenergy solvated protein fluctuations and underdamped intramolecular vibrations, respectively^{14}. Moreover, evidence is mounting that the interaction between excitons and underdamped vibrations whose energies commensurate exciton splittings may be at the heart of the coherence beating probed in twodimensional (2D) photon echo spectroscopy^{17,18,19,20,21,22,23}. Although some insights into the importance of such resonances can be gained from Förster theory^{24}, the wider implications for optimal spatiotemporal distribution of energy^{19,25,26}, for modulation of exciton coherences^{19,20,21,22} and for collective pigment motion dynamics^{23} have just recently started to be clarified.
The current state of the debate then suggests that a conceptual advance in understanding nontrivial quantum phenomena assisting excitation transport could emerge precisely from investigating nonclassical features of the molecular motions and phonon environments that play such a key role. Techniques able to manipulate vibrational states^{27,28} and probe their quantum properties^{29,30} may indeed provide the experimental platform to address this issue. Here we investigate this question in a prototype dimer ubiquitous in lightharvesting antennae of cyanobacteria^{31}, cryptophyte algae^{32,33} and higher plants^{34,35,36} and show that commensurate energies of exciton splitting and underdamped highenergy vibrations allows exciton–vibration dynamics to induce and harness nonclassical fluctuations of collective pigment motions for efficient energy transfer. Negative values of the Mandel Qparameter^{37} indicating subPoissonian phonon occupation fluctuations and, correspondingly, negative regions in a regularized quasiprobability P distribution in phase space^{38,39} unambiguously preclude any classical description of such fluctuations or its correlations with transport. Our results show a potential functional relevance of nonclassicality of molecular fluctuations for exciton transport and therefore provide a framework to investigate similar nontrivial quantum phenomena in the large variety of biomolecular transport^{40,41}, photoreception^{42} and chemical sensing processes^{43,44,45} that are known (or hypothesized) to be assisted by unequilibrated vibrational motion.
Results
Characterizing nonclassicality
The field of quantum optics has developed a solid framework to quantify the quantum properties of bosonic fields^{46}. It therefore provides excellent conceptual and quantitative tools to investigate nonclassicality of the harmonic vibrational degrees of freedom of interest in this work. From the perspective of quantum optics, quantum behaviour with no classical counterpart—that is, nonclassicality—arises if the state of the system of interest cannot be expressed as a statistical mixture of coherent states defining a valid probability measure^{47}. This then leads to nonpositive values of a phase–space quasiprobability distribution such as the Glauber–Sudarshan P(α)function^{47}.
where χ(ξ) is the characteristic function of the bosonic quantum state. However, highly singular behaviour of P(α) can make its characterization challenging both theoretically and experimentally. To overcome this, verification of the nonclassicality of a quantum state can be performed by constructing a regularized distribution P_{w}(α) as the Fourier transform of a filtered quantum characteristic function χ_{w}(ξ)^{38,39} as explained in the Methods section. Negativities in this regularized distribution are necessary and sufficient condition of quantum behaviour with no classical analogue^{38} and offer a significant advantage over other distributions, such as the Wigner distribution, which can be positive for quantum states that are truly nonclassical.
As an alternative to phase–space distributions, signatures of nonclassicality can be observed in the fluctuations of the bosonic field. For a single mode, negative values of the Mandel’s Q–parameter^{37} can be a signature of nonclassical behaviour. It characterizes the departure of the occupation number distribution P(n) from Poissonian statistics through the inequality
where and denote the first and second moments of the bosonic number operator , respectively. Vanishing Q indicates Poissonian number statistics where the mean of equals its variance as it is characteristic of classical wavelike behaviour—that is, a coherent state of light. For a chaotic thermal state, one finds that Q=>0 indicating that particles are ‘bunched’. A Fock state is characterized by Q<0 indicating that particle occupation is restricted to a particular level. Inequalities involving occupation probabilities of nearest number states can similarly witness nonclassical occupation fluctuations^{48}.
In what follows we use the above framework to investigate the nonclassical behaviour of the vibrational motions that drive excitation dynamics in prototype dimers present in a variety of antennae proteins of photosynthetic systems.
Prototype dimers and collective motion
We consider a prototype dimer where each chromophore has an excited electronic state of energy ε_{i} strongly coupled to a quantized vibrational mode of frequency ω_{vib} much larger than the thermal energy scale K_{B}T and described by the bare Hamiltonians H_{el}=∑_{i=1,2}ε_{i}σ_{i}^{+}σ_{i}^{−} and H_{vib}=ω_{vib}(b_{1}^{†}b_{1}+b_{2}^{†}b_{2}), respectively. Interchromophore coupling is generated by dipole–dipole interactions of the form H_{d−d}=V(σ_{1}^{+}σ_{2}^{−}+σ_{2}^{+}σ_{1}^{−}). The electronic excited states interact with their local vibrational environments with strength g, linearly displacing the corresponding mode coordinate, H_{el−vib}=g∑_{i=1,2}σ_{i}^{+}σ_{i}^{−}(b_{i}^{†}+b_{i}). In the above b_{i}^{†}(b_{i}) creates (annihilates) a phonon of the vibrational mode of chromophore i, while σ_{i}± creates or destroys an electronic excitation at site i. The eigenstates X› and Y› of H_{el}+H_{d−d} denote exciton states with energy splitting given by ΔE=((Δε)^{2}+4V^{2})^{1/2} and Δε=ε_{1}−ε_{2}. Transformation into collective mode coordinates shows that centre of mass mode b^{(†)}_{cm}=(b_{1}^{(†)}+b_{2}^{(†)})/ decouples from the electronic degrees of freedom and that only the mode corresponding to the relative displacement mode with bosonic operators
couples to the excitonic system. It is the nonclassical properties of this collective mode that we investigate. In collective coordinates, the effective exciton–vibration Hamiltonian then reads
with σ_{z}=σ_{2}^{+}σ_{2}^{−}−σ_{1}^{+}σ_{1}^{−} and σ_{x}=(σ_{1}^{+}σ_{2}^{−}+σ_{2}^{+}σ_{1}^{−}). Tiwary et al.^{23} have recently pointed out that 2D spectroscopy can probe the involvement of these anticorrelated, relative displacement motions in electronic dynamics. From now on and for simplicity, we denominate this relative displacement mode as the collective mode.
We are interested in dimers that satisfy ΔE~ω_{vib}>g>V where the effects of underdamped highenergy vibrational motions are expected to be most important^{19,31,36}. Several natural lightharvesting antennae include pairs of chromophores that clearly fall in this regime. Two important examples of such dimers are illustrated in Fig. 1a,b and correspond to the central PEB (phycoeritrobilin)_{50c}–PEB_{50d} dimer in the cryptophyte antennae PE545 (ref. 33) and a Chl_{b601}Chl_{a602} pair in the lightharvesting complex II (LHCII) complex of higher plants^{36}; both corresponding to systems that have exhibited coherence beating in 2D spectroscopy^{3,4,49}. Importantly, in each case, the dimer considered contributes to an important energy transfer pathway towards exit sites^{3,33}, suggesting that the the phenomena we discuss will have an effect in the performance of the whole complex. Moreover, synthetic versions of such prototype dimers could be available^{50}. Most remarkably, LHCII is likely the most abundant lightharvesting complex on Earth^{35}, while cryptophyte antennae such as PE545 are ecologically important as they support photosynthesis under extreme lowlight conditions^{51,52}. From this perspective, the dimers of interest are exceptionally relevant biomolecular prototypes. Spectroscopy studies indicate that these dimers are subject to a structured exciton–phonon interaction as considered in our model. For the PEB_{50} dimer, the intramolecular mode of interest has frequency around 1,111 cm^{−1} (ref. 33), which compares with the frequency of the breathing mode of the tetrapyrrole^{53} (Carles Curutchet, personal communication). In the case of the Chl_{b−a} pair, it has been shown that a mode around 750 cm^{−1} is coupled to the electronic dynamics^{36} and this energy is close to the frequency of inplane deformations of the pyrrole^{54}. Furthermore, vibrational dephasing in chromophores^{55} and in other systems such as photoreceptors^{56} is known to be of the order of picoseconds. Some aspects of the influence of nonequilibrium vibrational motion in these specific dimers have been considered before^{19,57}; however, none of these studies have addressed the question of interest: can vibrationassisted transport exploit quantum phenomena that have no classical analogue?
Nonclassicality via coherent exciton–vibration dynamics
We first consider the quantum coherent dynamics associated to H_{ex−vib} to illustrate how nonclassical behaviour of the collective motion emerges out of an initial thermal phonon distribution and an excitonic state with no initial superpositions: , which in the basis of exciton–vibration states of the form X,n› (see Fig. 2a), becomes . Here n denotes the phonon occupation number of the relative displacement mode coupled to exciton dynamics (see Eq. 4), while P_{th}(n) denotes the thermal occupation of such level. The observables of interest are the population of the lowest excitonic state ρ_{YY}(t)=∑_{n}〈Y,nρ(t)Y,n›, the absolute value of the coherence ρ_{X0−Y0}(t)=〈X,0ρ(t)Y,0›, which denotes the interexciton coherence in the ground state of the collective vibrational mode, and the nonclassicality given by negative values Q(t) and corresponding negativities in the regularized quasiprobability distribution P_{w}(α). Hamiltonian evolution generates coherent transitions from states X,n› to Y,n+1› (see Fig. 2a) with a rate f that depends on the exciton delocalization (V/Δε), the coupling to the mode g and the phonon occupation n—that is, . Since ω_{vib}≫K_{B}T the ground state of the collective mode is largely populated, such that the Hamiltonian evolution of the initial state is dominated by the evolution of the state X,0›. This implies that the energetically close exciton–vibration state Y,1› becomes coherently populated at a rate , leading to the oscillatory pattern observed in the probability of occupation ρ_{YY}(t) as illustrated in Fig. 2b. The lowfrequency oscillations of the dynamics of ρ_{YY}(t) cannot be assigned to the exciton or the vibrational degrees of freedom alone as expected from quantum coherent evolution of the excitonpluseffective mode system. For instance, if the mode occupation is restricted to at most n=1, the period of the amplitude of ρ_{YY}(t) is given approximately by the inverse of
with α^{2}=2g^{2}+ and (2g^{2}4V^{2}/α^{2}ΔE^{2})<<1. Coherent exciton population transfer is accompanied by beating of the interexciton coherence ρ_{X0−Y0}(t), with the main amplitude modulated by the same lowfrequency oscillations of ρ_{YY}(t) and a superimposed fast oscillatory component of frequency close to ω_{vib} (see Fig. 2b). This fastdriving component arises from local oscillatory displacements: when V≃0 the time evolution of each local mode is determined by the displacement operator with amplitude (ref. 58). As the state Y,1› is coherently populated, the collective quantized mode is driven out of equilibrium towards a nonclassical state in which selective occupation of the first vibrational level takes place, thereby modulating occupation of higher levels. This manifests itself in subPoissonian phonon statistics as indicated by negative values of Q(t) shown in Fig. 3a. Similar phenomena have been described in the context of electron transport in a nanoelectromechanical system^{59,60}. Moreover, Fig. 3b shows that at times when Q(t) is negative—that is, t=0.2 ps—the regularized quasiprobability distribution P_{w}(α) at this time exhibits negativities, thereby ruling out any classical description of the same phenomena. Interestingly, the nonclassical properties of the collective vibrational motion resemble nonclassicality of bosonic thermal states (completely incoherent states) that are excited by a single quanta^{39,61}. Importantly, such nonclassical behaviour of the vibrational motion arises only when the electronic interaction between pigments is finite. For comparison, Fig. 3c shows that if V=0, an electronic excitation drives the local underdamped vibration towards a thermal displaced state with superPoissonian statistics (Q(t)>0), which has an associated positive probability distribution in phase space as illustrated in Fig. 3d. In short, nonclassicality of the collective mode quasiresonant with the excitonic transition arises through the transient formation of exciton–vibration states.
Energy and coherence dynamics under thermal relaxation
We now investigate the dynamics of the exciton–vibration dimer when each local electronic excitation interacts additionally with a lowenergy thermal bath described by a continuous distribution of harmonic modes. The strength of this interaction is described by a Drude spectral density with associated reorganization energy λ and cutoff frequency Ω_{c}<K_{B}T as described in the Methods section. We consider the exciton and vibration parameters to the PEB_{50} dimer and investigate the trends as functions of the reorganization energy. As expected, the interplay between vibrationactivated dynamics and thermal fluctuations leads to two distinct regimes of energy transport as a function of λ. For our consideration of weak electronic coupling, the coherent transport regime is determined approximately by (λΩ_{c})^{1/2}≤2gV/Δ∈. Population of the lowlying exciton state is dominated by coherent transitions between excitoncollective mode states and the rapid, nonexponential growth of ρ_{YY}(t) in this regime can be traced back to coherent evolution from X,0› to Y,1›. At longer timescales thermal fluctuations induce incoherent transitions from X,0› to Y,0› with a rate proportional to (λΩ_{c})^{1/2}, thereby stabilizing population of ρ_{YY}(t) to a particular value as can be seen in Fig. 4d. This behaviour is illustrative of what is expected in the dimer Chl_{b−a} for which λ=37 cm^{−1} as obtained from^{40}. To confirm this we have computed the exciton–vibration dynamics with parameters of the Chl_{b−a} dimer, the results of which are shown in Supplementary Fig. S1. In contrast, for (λΩ_{c})^{1/2}>2gV/Δ∈ population transfer to ρ_{YY}(t) is incoherent. For the PEB_{50} dimer λ~110 cm^{−1} that place this dimer in this incoherent regime where ρ_{YY}(t) has a slow but continuous exponential rise reflecting the fact that thermal fluctuations inducing transitions from X,n› to Y,n› now have a large contribution to exciton transport. However, even in this regime, transfer to ρ_{YY}(t) is always more efficient with the quasiresonant mode than in the situations where only thermalbathinduced transitions are considered (see dashed lines in Fig. 4a,d,g). The underlying reason is that before vibrational relaxation takes place (around t=1 ps), the system is transiently evolving towards a thermal configuration of excitoncollective mode states. Hence, in both coherent and incoherent population transfer regimes, transfer to the lowest exciton state involves a transient, selective population of the first vibrational level of the collective mode. The transition from coherent to incoherent exciton population dynamics is then marked by the onset of energy dissipation of the exciton–vibration system as shown in Fig. 4c,f,i, where E(t)=Tr{H_{ex−vib}ρ(t)} has been depicted for different values of λ. While exciton population growth is nonexponential, energy dissipation into the thermal bath is transiently prevented as indicated by periods of positive slope of E(t) as happens in Fig. 4c,f. Quantification of the energy that is transiently ‘extracted’ from the lowenergy thermal bath can provide an interesting physical interpretation of the advantages of nonexponential exciton transfer in the framework of nonequilibrium thermodynamics^{62}.
For completeness, we present in Figure 4b,e,h how the beating patterns of the coherence ρ_{X0−Y0}(t) reveal the structured nature of the exciton–phonon interaction and witnesses whether there is coherent exciton–vibration evolution as it has been pointed out by recent studies^{21,22}. The frequency components of such oscillatory exciton coherences vary depending on the coupling to the thermal bath. In the coherent regime, as ρ_{X0−Y0}(t) follows exciton populations, the main amplitude is modulated by the same relevant energy difference between exciton–vibration states (see Fig. 4b,e). This behaviour is relevant for the parameter regime of the Chl_{b−a} dimer (see Supplementary Fig. S1). In contrast, for the PEB_{50} dimer, the shorttime oscillations of ρ_{X0−Y0}(t) (between t=0 and t=0.1 ps) arise from purely electronic correlations because of bathinduced renormalization of the electronic Hamiltonian^{63}. This exciton coherence retains the superimposed driving at a frequency ω_{vib} and is accompanied by nonclassicality as it will be described shortly, indicating that vibrational motion is still out of thermal equilibrium. The dynamical features presented in Fig. 4g,h agree with previous findings based on a perturbative approach^{19} and with the timescales of the exciton coherence beating reported for cryptophyte algae^{4,18}.
Nonclassicality under thermal relaxation
Interaction with the thermal environment would eventually lead to the emergence of classicality in longer timescales. However, in the picosecond timescale of interest, the collective mode exhibits periods of nonclassicality across a wide range of thermal bath couplings λ as indicated by subPossonian fluctuations with Q(t)<0 in Fig. 5a–c and the corresponding negativities in the distributions P_{w}(α) shown in Fig. 5d–f. This survival of nonclassicality is concomitant with a slow decay of the exciton–vibration coherence ρ_{X0,Y1}(t) (not shown). Nonclassical behaviour of collective fluctuations are then expected for the parameters of both the PEB_{50} dimer for which λ=110 cm^{−1} and the Chl_{b−a} dimer for which λ=37 cm^{−1}. The nonclassical fluctuations predicted by Q(t) also agree with those witnessed by a parameter quantifying correlations between nearestneighbours’ occupations^{48}, which we present in Supplementary Fig. S2. As expected, the maximum nonclassicality indicated by the most negative value of Q(t) decreases for larger reorganization energies. Nonetheless, the time average of these nonclassicalities is not a monotonic function of λ. For moderate values of λ, the collective mode spends longer periods in states with nonclassical fluctuations—that is, periods for which Q(t)<0 as seen in Fig. 5b, thereby stabilizing nonclassicality at a particular level. This subpicosecond stabilization of nonclassicality is expected in the regime of the Chl_{b−a} dimers as illustrated in Supplementary Fig. S1.
Functional role of nonclassicality
Nonclassical fluctuations of collective motions correlate to exciton population transfer. In order to demonstrate this, we investigate quantitative relations between nonclassicality and exciton energy transport by considering relevant integrated averages in the timescale of the Hamiltonian evolution of the exciton–vibration system denoted by τ. For the parameters of the PEB_{50}, this timescale is about half a picosecond and is comparable to the timescale in which excitation energy would be distributed away to other chromophores or to a trapping state. The timeintegrated averages over τ are defined as: 〈F[ρ(t)]›_{τ}=(1/τ)∫_{0}^{τ}dt F[ρ(t)], where F[ρ(t)] corresponds to the exciton population ρ_{YY}(t) and the nonclassicality of the underdamped collective mode through periods of subPoissonian statistics Q(t)Θ[−Q(t)] as functions of the coupling to the bath λ. As shown in Fig. 6, the average exciton population and nonclassicality follow a similar nonmonotonic trend as a function of the coupling to the thermal bath, indicating a direct quantitative relation between efficient energy transfer in the timescale τ and the degree of nonclassicality. The appearance of a maximal point in the average nonclassicality as a function of the systembath coupling indicates that the average quantum response of the collective anticorrelated motion to the impulsive electronic excitation is optimal for a small amount of thermal noise.
It is worth emphasizing that the above functional role of nonclassicality holds for vibrationassisted transport where the highenergy intramolecular modes considered are quasiresonant with the excitonic energy splitting. When vibrational motions are significantly detuned with the bare exciton transition ΔE, transport is dominated by the thermal background and no selective population of the state Y,1› takes place; hence, periods of subPoissonian fluctuations vanish. To illustrate the difference with the offresonance case, Fig. 7 shows the same timeintegrated averages as in Fig. 6, with the electronic parameters of the PEB_{50} dimer but now considering an intramolecular vibration of frequency ω_{vib}=1,520 cm^{−1} significantly detuned from ΔE. In this case, thermally activated transport (see dashed line in Fig. 7) and vibrationassisted transport (solid line in Fig. 7) are practically indistinguishable and the average Q(t) is positive with a value near zero as expected for a thermal distribution of a highenergy harmonic mode.
The degree of purity of the initial exciton state is also important in enabling and harnessing nonclassical fluctuations of the collective mode. One therefore should expect that statistical mixtures of excitons with finite purity can still trigger such nonclassical response. To illustrate this, we now consider mixed initial states of the form ρ(t_{0})=ρ_{ex}⊗ρ^{th}_{vib} where with 1/2≤r≤1. The associated linear entropy quantifying the mixedness of the initial exciton states is given by S_{L}=2r(r−1). The timeaveraged nonclassicality (Fig. 8a) and average population transfer (Fig. 8b) follow similar decreasing, yet nonzero, monotonic trends for mixed states. These results suggest that nonclassical vibrational motion can be prompted and exploited even under incoherent conditions creating statistical mixture of excitons^{64}. The trends presented in Figs 6, 8 constitute theoretical evidence of direct quantitative correlations between nonclassicality and exciton population transport in a relevant subpicosecond timescale and therefore illustrate a functional role for quantum phenomena with no classical counterpart in a prototype lightharvesting system. Our results remain valid when the picoseconddephasing rate of the vibrational motion is included in the dynamics as can be seen in Supplementary Fig. S3.
Discussion
Lightharvesting complexes are fundamental components of the photosynthetic machinery on Earth. While there has been an extraordinary advance in the techniques to probe these systems at ultrafast timescales, it is conceptually unclear what truly quantum features with no classical counterpart these systems may exhibit and exploit for efficient energy transport^{65}. Besides depending on longrange electronic interactions, excitation energy distribution is fundamentally affected by the molecular motions and harmonic (or in cases anharmonic) environments that modulate the electronic dynamics. We therefore put forward the idea that, precisely, investigation of nonclassical phenomena associated to such molecular motions can pave the way towards understanding which nontrivial quantum phenomena can have an impact on efficient energy distribution and trapping. Within this line of thought, we investigate nonclassical behaviour in a prototype dimer that conveys fundamental physical principles of vibrationassisted transport in a variety of lightharvesting antennae. We demonstrate that quasiresonances between excitonic transitions and underdamped highenergy intramolecular vibrations can trigger and harness nontrivial quantum behaviour of collective pigment motions that are initially in a thermal, fully incoherent, state. In this scenario, correlations between nonclassical fluctuations of collective pigment motions and efficient population of the target exciton state are found. Negative values both of the Mandel Qparameter and of the quasiprobability distribution P_{w}(α) of the collective motion assure that no classical distribution can describe the same behaviour. These nontrivial quantum phenomena are predicted for a variety of initial excitation conditions including statistical mixtures of excitons indicating that such nonclassicality can be activated even under incoherent input of photoexcitations. Transient coherent ultrafast phonon spectroscopy^{29}, which is sensitive to lowphonon populations of highenergy vibrations^{30}, may provide an interesting experimental approach to probe the phenomena we describe.
The prototype exciton–vibration dimers here investigated are representative of interbandlike transfer pathways present in the majority of lightharvesting complexes. For instance, in the LHCII complex, the considered dimer contributes to the fastest component of the Chl_{b}→Chl_{a} interband transfer pathway that directs excitation energy towards exit sites^{66}. The demonstration that nonclassicality is concomitant with efficient vibrationassisted transfer in these dimers therefore suggests that nonclassical phenomena will have a contribution to the efficiency of the whole complex. A rigorous quantitative estimation of such contribution requires both a careful extension of our formalism to quantify these features in a multimodal system (as likely other nonequilibrated vibrational motion will be involved) and a careful weighting of the vibrationassisted processes in the overall spatiotemporal distribution of energy.
The framework we propose can also be applied to gain insights into the nonclassical response of vibrational motion in a variety of transport^{40,41} and sensing processes in biomolecules^{42,43}. Of particular interest are charge transfer in reaction centres^{40} and isomerization of photoreceptors^{42} where specific intramolecular vibrational motions are known to be driven out of thermal equilibrium during the lightinitiated electronic dynamics. It will also be interesting to use this framework to understand possible nontrivial quantum behaviour of molecular motions in chemical sensors^{43} that are conjectured to operate through weak electronic interactions to sense molecular vibrations of the order of a thousand wavenumbers^{44,45}.
We have also illustrated how in our prototype dimer with biologically relevant parameters, exciton–vibration dynamics can lead to nonexponential excitonic energy distribution whereby dissipation into a lowenergy thermal bath can be transiently prevented. From this view, coherent vibrational motions that do not relax quickly and whose fluctuations cannot be described classically may be seen as an internal quantum mechanism controlling energy distribution and storage. Further insights into the advantage of these nontrivial quantum behaviour may therefore be gained in a thermodynamic framework^{62,67}.
In conclusion, we have provided theoretical evidence that vibrationassisted exciton transport in prototype dimers, representative of interbandlike transitions in a variety of photosythetic lightharvesting antennae, can exploit nontrivial quantum phenomena that cannot be reproduced by any classical counterpart, namely, nonclassical fluctuations of collective pigment motions. Given that a variety of transport^{40,41} and sensing phenomena^{42} in biomolecules are known to involve nonequilibrium vibrational motion, our findings have broad implications for the field of quantum effects in biology as they suggest that investigating the nonclassical nature of molecular fluctuations harnessed in these processes could be the key to reveal a role for truly nontrivial quantum features.
Methods
Open quantum system dynamics
To accurately account for the effects of the lowenergy thermal bath, we have adopted a hierarchical expansion of the exciton–vibration dynamics^{68,69,70}. We split the highenergy mode from the bath of harmonic oscillators and explicitly treat the quantum interactions between electronic excitations and these modes of frequency ω_{vib} by including it within the definition of the system, H_{ex−vib}=H_{ex}⊗11_{vib}+11_{ex}⊗H_{vib}+H_{ex−vib}. The electronic operators then couple to the remaining vibrational modes H_{I}=∑_{i,k}g_{i}(σ_{i}^{+}σ_{i}^{−}⊗11_{vib})(b^{†}_{k}+b_{k}). This approach allows the effects of the lowenergy thermal bath on the exciton–vibration dynamics to be accurately accounted for. Truncating the quantized mode at Fock level M=6 adequately describes both reduced dynamics of the collective quantized mode and the electronic dynamics of the prototype dimer at room temperature. A spectral density of the form J(ω)=2λΩ_{c}ω/(Ω_{c}^{2}+ω^{2}) is assumed, where λ and Ω_{c} are the reorganization energy and cutoff frequency, respectively. Supplementary Note 1 furnishes further details of the hierarchical expansion of exciton–vibration dynamics. Converged dynamics are obtained by terminating the hierarchical expansion at level N=11 and including just the K=0 Matsubara term. No additional Matsubara terms are necessary as Ω_{c}<K_{B}T. For completeness, we present dynamics including the K=1 term in Supplementary Fig. S4.
Regularized quasiprobability distributions
The quasiprobability distribution we calculate is a regularized version of the Prepresentation P_{w}(α)=(1/π^{2})∫d^{2}ξ e^{αξ*−α}^{*ξ}χ(ξ)Ω_{w}(ξ), where the quantum characteristic function χ(ξ)=\text{Tr{}{e}^{\xi {b}_{\text{rd}}^{\u2020}\xi *{b}_{\text{rd}}}{\rho}_{\text{vib}}\} of the reduced state the vibration ρ_{vib} is reconstructed in the truncated Fock basis . Ω_{w}(ξ) is a nonclassicality filter^{38} taken as the triangular function with w=5, which fulfils the necessary condition of a filter such that P_{w}(ξ) detects nonclassicality. More details are given in Supplementary Note 2.
Additional information
How to cite this article: O’Reilly, E. J. and OlayaCastro, A. Nonclassicality of the molecular vibrations assisting exciton energy transfer at room temperature. Nat. Commun. 5:3012 doi: 10.1038/ncomms4012 (2014).
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Acknowledgements
We thank Greg Scholes and Rienk van Grondelle for discussions. We are also grateful to Carles Curuchet for a preliminary analysis on the specific molecular origins of vibrational modes in the PE545 complex and to Avinash Kolli for his support on numerical calculations in the initial stages of this project. Financial support from the Engineering and Physical Sciences Research Council (EPSRC UK) Grant EP/G005222/1 and from the EU FP7 Project PAPETS (GA 323901)is gratefully acknowledged.
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E.J.O. performed the calculations. A.O.C. conceived, designed and supervised the research. Both authors analysed and discussed the results and cowrote the manuscript.
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O’Reilly, E., OlayaCastro, A. Nonclassicality of the molecular vibrations assisting exciton energy transfer at room temperature. Nat Commun 5, 3012 (2014). https://doi.org/10.1038/ncomms4012
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DOI: https://doi.org/10.1038/ncomms4012
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