Characterizing non-classicality

The field of quantum optics has developed a solid framework to quantify the quantum properties of bosonic fields46. It therefore provides excellent conceptual and quantitative tools to investigate non-classicality 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, non-classicality—arises if the state of the system of interest cannot be expressed as a statistical mixture of coherent states defining a valid probability measure47. This then leads to non-positive values of a phase–space quasi-probability distribution such as the Glauber–Sudarshan P(α)-function47.

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 non-classicality 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 analogue38 and offer a significant advantage over other distributions, such as the Wigner distribution, which can be positive for quantum states that are truly non-classical.

As an alternative to phase–space distributions, signatures of non-classicality can be observed in the fluctuations of the bosonic field. For a single mode, negative values of the Mandel’s Q–parameter37 can be a signature of non-classical 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 wave-like 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 non-classical occupation fluctuations48.

In what follows we use the above framework to investigate the non-classical 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. Inter-chromophore 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+4V2)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 non-classical 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 high-energy vibrational motions are expected to be most important19,31,36. Several natural light-harvesting 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 light-harvesting complex II (LHCII) complex of higher plants36; both corresponding to systems that have exhibited coherence beating in 2D spectroscopy3,4,49. Importantly, in each case, the dimer considered contributes to an important energy transfer pathway towards exit sites3,33, 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 available50. Most remarkably, LHCII is likely the most abundant light-harvesting complex on Earth35, while cryptophyte antennae such as PE545 are ecologically important as they support photosynthesis under extreme low-light conditions51,52. 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 tetrapyrrole53 (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 dynamics36 and this energy is close to the frequency of in-plane deformations of the pyrrole54. Furthermore, vibrational dephasing in chromophores55 and in other systems such as photoreceptors56 is known to be of the order of picoseconds. Some aspects of the influence of non-equilibrium vibrational motion in these specific dimers have been considered before19,57; however, none of these studies have addressed the question of interest: can vibration-assisted transport exploit quantum phenomena that have no classical analogue?

Figure 1: Prototype dimers. (a,b) Cryptophyte antennae phycoerythrin 545 (PE545) and LHCII present in higher plants have pairs of pigments whose electronic and vibrational parameters fall in the regime of our vibration-assisted transport model. (a) Representation of the pigments and protein environment of a PE545 complex of Rhodomonas CS24 (Protein Data Bank ID code 1XG0, ref. 32). The central PEB dimer pigments PEB 50c and PEB 50d are highlighted in red and green, respectively. For this PEB 50 dimer, there is an uncertainty in the value of the energy gap32,33. We take parameters from refs 32, 33 such that Δε=1,042 cm−1 and V=92 cm−1, so ΔE=1,058.2 cm−1 being quasi-resonant with an intramolecular mode of frequency ω vib =1,111 cm−1. The strength of linear coupling to this mode is g=ω vib (0.0578)1/2=267.1 cm−1. (b) Representation of the LHCII antennae of Spinacia oleracea (Protein Data Bank ID code 1RWT, ref. 34). Several pairs of close Chl b -Chl a (red-green) chlorophylls satisfy the conditions of our model. In particular, we consider the Chl b601 -Chl a602 pair for which Δε=661 cm−1 and V=−47.1 cm−1, resulting in ΔE=667.7 cm−1 (ref. 66). An intramolecular vibrational mode of frequency ω vib =742.0 cm−1 is close to this energy gap and each chromophore couples to this mode with strength g=ω vib (0.03942)1/2=147.3 cm−1 as obtained from (ref. 40). Scale bar: 1 nm. Full size image

Non-classicality via coherent exciton–vibration dynamics

We first consider the quantum coherent dynamics associated to H ex−vib to illustrate how non-classical 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,n|ρ(t)|Y,n›, the absolute value of the coherence ρ X0−Y0 (t)=〈X,0|ρ(t)|Y,0›, which denotes the inter-exciton coherence in the ground state of the collective vibrational mode, and the non-classicality given by negative values Q(t) and corresponding negativities in the regularized quasi-probability 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 low-frequency 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 exciton-plus-effective 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

Figure 2: Exciton-collective mode states and free exciton dynamics. (a) The energy levels of the exciton-collective mode states used to describe energy transfer . The red arrow denotes population transfer from to . (b) Quantum coherent dynamics of the PEB 50 dimer in PE545 illustrating population of the lowest exciton ρ YY (t) (thick blue curve) and the inter-exciton coherence in the ground-state of the collective mode |ρ X0−Y0 (t)| (thin red curve). Full size image

with α2=2g2+ and (2g24V2/α2ΔE2)<<1. Coherent exciton population transfer is accompanied by beating of the inter-exciton coherence |ρ X0−Y0 (t)|, with the main amplitude modulated by the same low-frequency oscillations of ρ YY (t) and a superimposed fast oscillatory component of frequency close to ω vib (see Fig. 2b). This fast-driving 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 non-classical state in which selective occupation of the first vibrational level takes place, thereby modulating occupation of higher levels. This manifests itself in sub-Poissonian 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 system59,60. Moreover, Fig. 3b shows that at times when Q(t) is negative—that is, t=0.2 ps—the regularized quasi-probability distribution P w (α) at this time exhibits negativities, thereby ruling out any classical description of the same phenomena. Interestingly, the non-classical properties of the collective vibrational motion resemble non-classicality of bosonic thermal states (completely incoherent states) that are excited by a single quanta39,61. Importantly, such non-classical 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 super-Poissonian statistics (Q(t)>0), which has an associated positive probability distribution in phase space as illustrated in Fig. 3d. In short, non-classicality of the collective mode quasi-resonant with the excitonic transition arises through the transient formation of exciton–vibration states.

Figure 3: Non-classicality of collective and local vibrational modes. (a) Mandel Q-parameter of the relative displacement mode of the considered PEB 50 dimer from PE545 when a biological electronic coupling is considered. Shaded regions denote times of non-classicality. (b) The associated regularized quasi-probability distribution P w (α) at t=0.2 ps. Shaded regions denote areas of negative probability. (c) Mandel Q-parameter of the intramolecular high-energy vibration when dipole coupling is zero and (d) the associated regularized quasi-probability distributions P w (α) at t=0.2 ps. Full size image

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 low-energy 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 vibration-activated 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≤2g|V|/Δ∈. Population of the low-lying exciton state is dominated by coherent transitions between exciton-collective mode states and the rapid, non-exponential 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 from40. 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>2g|V|/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 quasi-resonant mode than in the situations where only thermal-bath-induced 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 exciton-collective 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 non-exponential, 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 low-energy thermal bath can provide an interesting physical interpretation of the advantages of non-exponential exciton transfer in the framework of non-equilibrium thermodynamics62.

Figure 4: Energy and coherence evolution under thermal relaxation. The dynamics of ρ YY (t) with (thick blue curve) and without (dashed curve) coupling to vibration (top row), |ρ X0,Y0 (t)| (middle row) and energy of the exciton vibration system E(t)=Tr{H ex−vib ρ(t)} (bottom row) for the exciton–vibration parameters of the PEB 50 dimer and three interaction strengths to the low-energy thermal bath: (a–c): λ=6 cm−1, (d–f) λ=35 cm−1 and (g–i): λ=110 cm−1. Initial energy E(0), displayed as dashed line and times where E(t)>E(0) are shaded. Full size image

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 studies21,22. 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 short-time oscillations of ρ X0−Y0 (t) (between t=0 and t=0.1 ps) arise from purely electronic correlations because of bath-induced renormalization of the electronic Hamiltonian63. This exciton coherence retains the superimposed driving at a frequency ω vib and is accompanied by non-classicality 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 approach19 and with the timescales of the exciton coherence beating reported for cryptophyte algae4,18.

Non-classicality 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 non-classicality across a wide range of thermal bath couplings λ as indicated by sub-Possonian 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 non-classicality is concomitant with a slow decay of the exciton–vibration coherence ρ X0,Y1 (t) (not shown). Non-classical 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 non-classical fluctuations predicted by Q(t) also agree with those witnessed by a parameter quantifying correlations between nearest-neighbours’ occupations48, which we present in Supplementary Fig. S2. As expected, the maximum non-classicality indicated by the most negative value of Q(t) decreases for larger reorganization energies. Nonetheless, the time average of these non-classicalities is not a monotonic function of λ. For moderate values of λ, the collective mode spends longer periods in states with non-classical fluctuations—that is, periods for which Q(t)<0 as seen in Fig. 5b, thereby stabilizing non-classicality at a particular level. This sub-picosecond stabilization of non-classicality is expected in the regime of the Chl b−a dimers as illustrated in Supplementary Fig. S1.

Figure 5: Non-classicality of collective motions under thermal relaxation. (a–c) Dynamics of the Mandel Q-parameter for λ=6, 35, and 110 cm−1, respectively. Shaded regions denote times of non-classicality. (d–f) Regularized distribution P w (α) at t=0.2 ps for each corresponding value of λ. Shaded regions denote areas of negative probability. Full size image

Functional role of non-classicality

Non-classical fluctuations of collective motions correlate to exciton population transfer. In order to demonstrate this, we investigate quantitative relations between non-classicality 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 time-integrated averages over τ are defined as: 〈F[ρ(t)]› τ =(1/τ)∫ 0 τdt F[ρ(t)], where F[ρ(t)] corresponds to the exciton population ρ YY (t) and the non-classicality of the underdamped collective mode through periods of sub-Poissonian 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 non-classicality follow a similar non-monotonic 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 non-classicality. The appearance of a maximal point in the average non-classicality as a function of the system-bath 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.

Figure 6: Correlations between non-classicality and exciton populations. Time-integrated averages of exciton population ρ YY (t) and non-classicality as quantified by Q(t)Θ[−Q(t)] (blue and green, respectively) as functions of coupling to the thermal background by fixing environment cutoff frequency Ω c =100 cm−1 and varying reorganization energy λ with exciton–vibration parameters corresponding to the PEB 50 dimer. Full size image

It is worth emphasizing that the above functional role of non-classicality holds for vibration-assisted transport where the high-energy intramolecular modes considered are quasi-resonant 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 sub-Poissonian fluctuations vanish. To illustrate the difference with the off-resonance case, Fig. 7 shows the same time-integrated 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 vibration-assisted 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 high-energy harmonic mode.

Figure 7: Effects of off-resonance vibrations. Time-integrated averages as in Fig. 6 with electronic parameters of the PEB 50 dimer but now considering an intramolecular vibration which is ω vib =1,520 cm−1—that is, significantly detuned from the energy gap ΔE=1,058.2 cm−1 and similar coupling strength g=ω vib (0.0265)1/2≈247 cm−1. Parameters obtained from ref. 33. The dashed curve shows time average of ρ YY (t) when no intramolecular vibration is considered and transport results only because of the thermal background. Full size image

The degree of purity of the initial exciton state is also important in enabling and harnessing non-classical fluctuations of the collective mode. One therefore should expect that statistical mixtures of excitons with finite purity can still trigger such non-classical response. To illustrate this, we now consider mixed initial states of the form ρ(t 0 )=ρ ex ⊗ρth 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 time-averaged non-classicality (Fig. 8a) and average population transfer (Fig. 8b) follow similar decreasing, yet non-zero, monotonic trends for mixed states. These results suggest that non-classical vibrational motion can be prompted and exploited even under incoherent conditions creating statistical mixture of excitons64. The trends presented in Figs 6, 8 constitute theoretical evidence of direct quantitative correlations between non-classicality and exciton population transport in a relevant sub-picosecond timescale and therefore illustrate a functional role for quantum phenomena with no classical counterpart in a prototype light-harvesting system. Our results remain valid when the picosecond-dephasing rate of the vibrational motion is included in the dynamics as can be seen in Supplementary Fig. S3.