Probing quantum features of photosynthetic organisms

Recent experiments have demonstrated strong coupling between living bacteria and light. Here we propose a scheme capable of revealing non-classical features of the bacteria (quantum discord of light-bacteria correlations) without exact modelling of the organisms and their interactions with external world. The scheme puts the bacteria in a role of mediators of quantum entanglement between otherwise non-interacting probing light modes. We then propose a plausible model of this experiment, using recently achieved parameters, demonstrating the feasibility of the scheme. Within this model we find that the steady state entanglement between the probes, which does not depend on the initial conditions, is accompanied by entanglement between the probes and bacteria, and provides independent evidence of the strong coupling between them.

There is no a priori limit on the complexity, size or mass of objects to which quantum theory is applicable. Yet, whether or not the physical configuration of macroscopic systems could showcase quantum coherences has been the subject of a long-standing debate. The pioneers of quantum theory, such as Schrödinger [1] and Bohr [2], wondered about whether there might be limitations to living systems obeying the laws of quantum theory. Wigner even claimed that their behaviour violates unitarity [3].
A striking way to counter such claims on the implausibility of macroscopic quantum coherence would be the successful preparation of quantum superposition states of living objects. A direct route towards such goal is provided by matter-wave interferometers, which have already been instrumental in observing quantum interference from complex molecules [4], and are believed to hold the potential to successfully show similar results for objects as large as viruses in the near future.
However, other possibilities exist that do not make use of interferometric approaches. An instance of such alternatives is to interact a living object with a quantum system in order to generate quantum correlations. Should such correlations be as strong as entanglement, measuring the quantum system in a suitable basis could project the living object into a quantum superposition. However, requesting the establishment of entanglement is, in general, not necessary as the presence of quantum discord, that is a weaker form of quantum correlations, would already provide evidence that the Hilbert space spanned by the living object must contain quantum superposition states [5][6][7][8][9]. For example, by operating on the quantum system alone one could remotely prepare quantum coherence in the living object [10].
A promising step in this direction, demonstrating strong coupling between living bacteria and optical fields and suggesting the existence of entanglement between them [11], has recently been realised [12]. However, the experimental results reported in Ref. [12] can as well be explained by a fully classical model [11,13], which calls loud for the design of a configuration with more conclusive interpretations.
In this paper we make a proposal in such a direction by designing a thought experiment that does not require any direct operation on the bacteria nor does it rely on any knowledge about their interaction with light. We then propose a plausible model of such interactions and demonstrate within this model that steady-state entanglement between the modes of the optical probes is always accompanied by light-bacteria entanglement, which is in turn empowered by the strong coupling between such systems. Our proposal thus delineates a path towards the inference of non-classicality of living organisms through their ability to establish quantum correlations with easily accessed probing systems.
The thought experiment and its model. Consider the setup presented in Fig. 1. The bacteria are inside a driven single-sided multimode Fabry-Perot cavity where they interact independently with a few cavity modes. It is crucial for the method that these modes do not interact FIG. 1. We consider a driven single-sided multimode Fabry-Perot cavity embedding green sulphur bacteria. Here, R1 is the input mirror, while R2 is a perfectly reflecting end mirror. A few cavity modes individually interact with the bacteria, but not with each other. Both bacteria and cavity modes are open systems. In particular, the interaction between bacteria and their environment results in the energy decay rate 2γ. The m th cavity field mode experiences energy dissipation at a rate 2κm.
directly, but only via the bacteria. This can be realised in practice by choosing cavity modes that are orthogonally polarised or sufficiently distant in frequency. Both bacteria and cavity are treated as open systems, each coupled to its own local environment.
Under such general conditions, if one starts from a global product state of the system, intermodal entanglement cannot be established if, at all times, there is no quantum discord between cavity modes and bacteria [14]. Conversely, any observed intermodal entanglement witnesses the presence of quantum discord. We stress that, for such a revelation of non-classicality of the bacteria, their state does not have to be measured and their interaction with the optical modes can remain unspecified. This embodies the methodological pillar of our proposal for the probing of quantum features in living organisms.
In order to make such proposal more concrete, we now study a specific model for the energy of the bacteria and their interactions with light in order to demonstrate that there is observable intermodal entanglement. We consider a photosynthetic bacterium, Chlorobaculum tepidum, that is able to survive in extreme environments with almost no light [15]. Each bacterium, which is approximately 2µm × 500nm in size, contains 200 chlorosomes, each having 200, 000 bacteriochlorophyll c (BChl c) molecules. Such pigment molecules serve as excitons that can be coupled to light [12,16].
Following the proposal in Ref. [11], we model the pigment molecules of the bacteria as a collection of N twolevel atoms. For simplicity, we assume that all of them have the same frequency ω b and are coupled through a dipole-like mechanism to each light mode. This simple model was already shown to be capable of explaining the results of recent experiments [12]. For N 1, such collection of two-level systems can be approximated to a spin N/2 angular momentum. In the low-excitation approximation (which we will justify later), such angular momentum can be mapped into an effective harmonic oscillator through the use of the well-known Holstein-Primakoff transformation [17]. This allows us to cast the energy of the overall system as Here, m = 1, . . . , M is the label for the m th cavity mode, whose annihilation (creation) operator is denoted asâ m (â † m ) and having frequency ω m . Moreover,b andb † denote the bosonic annihilation and creation operators for the harmonic oscillator describing the bacteria, which is coupled to the m th cavity field at rate G m . The collective form of the coupling allows us to write G m = g m √ N with g m = ( µ·Ê) π c/n 2 r λ m ε 0 V m , where µ is the dipole moment of the two-level transition,Ê a unit vector in the direction of cavity electric field, n r refractive index of medium, and λ m wavelength of the m th cavity mode with volume V m [18]. The cavity is driven by a multimode laser, each mode having frequency Ω m , amplitude E m = 2P m κ m / Ω m , power P m , and amplitude decay rate κ m . It is important to notice that in Eq. (1) we have not invoked the rotating-wave approximation but actually retained the counter-rotating termsâ mb andâ † mb † . These cannot be ignored in the regime of strong coupling and we will show that they actually play a crucial role in our proposal.
We assume the local environment of bacteria to give rise to Markovian open-system dynamics, which is modelled as decay of the two-level systems in their environment. On the other hand, the cavity modes are interacting with the electromagnetic environment outside the cavity, resulting in decay rate of each mode. We treat these environments as independent, i.e. they affect their open systems locally. In this case, the dynamics of optical modes and bacteria can be written using the standard Langevin formulation in Heisenberg picture. This gives the following equations of motion, taking into account noise and damping terms coming from interactions with the local environmentṡ where γ is the amplitude decay rate of the bacterial system.F m andF b are operators describing independent zero-mean Gaussian noise affecting the m th cavity field and bacteria respectively. The only correlation functions between these noises are F [19,20]. We express the Langevin equations in terms of mode quadratures. In particular, by usingx m ≡ (â m +â † m )/ one gets a set of Langevin equations for the quadratures that can be written in a matrix equationu(t) = Ku(t) + l(t) with the vector u = ( Here, K is a square matrix with dimension 2(M + 1) describing the drift and l is a 2(M + 1) vector containing the noise and pumping terms (see Appendix A for explicit expressions). The solution to the Langevin equations is given by where W ± (t) = exp (±Kt).
As the Langevin equations are linear this dynamics preserves gaussian character of the initial state and hence our system is fully characterised by its covariance matrix. One can construct the covariance matrix as a function of time V (t) from Eq. (3) (cf. Appendix B). Time evolution of important quantities can then be calculated from the covariance matrix, e.g. entanglement and excitation number (cf. Appendix C and D). We shall only be interested in the steady state, which is guaranteed when all real parts of the eigenvalues of K are negative. In this case the covariance matrix satisfies Lyapunov-like equation where D = Diag[κ 1 , κ 1 , · · · , κ M , κ M , γ, γ]. Note that the steady-state covariance matrix does not depend on the initial conditions, i.e. V (0). Let us suppose that initially G m = 0, this can be done for example by choosing a cavity such that the optical modes supported and dipole moment of bacterial system are orthogonal or by filtering out polarisation in the direction of µ (e.g. [21]). The cavity modes and bacteria can thus be assumed to be initially uncorrelated. In what follows, we start with vacuum state for the cavity modes and ground state for bacteria. The initial state of the bacteria is justified by the fact that ω b k B T , even at room temperature. The dynamics is then started by rotating the polarisation of cavity modes, i.e. turning on G m .
Results. In order to maximise the free spectral range of the cavity, its size should be as small as possible.
In the present case the bacteria themselves are about half a micron long. We therefore place the bacteria (ω b = 2.5 × 10 15 Hz) in a single-sided Fabry-Perot cavity of length L = 567 nm (cf. Fig. 1) [12]. The refractive index due to aqueous bacterial solution embedded in the cavity is n r ≈ 1.33, which implies the frequency of the m th cavity mode ω m = mπc/n r L ≈ 1.25m × 10 15 Hz. The mirrors are engineered such that R 2 = 100% and R 1 = 5%. We assume the reflectivities are the same for all the optical modes giving κ m ≈ 3×10 14 Hz. The decay rate of the excitons can be approximated as γ = 1/2τ b where τ b = 2h/Γ b is the coherence time with Γ b the FWHM linewidth [22]. The FWHM linewidth is taken to be 130 meV [12], giving γ ≈ 7.8×10 12 Hz. We take all the spectral components of the driving laser to have the same power P m = 50mW and frequency Ω m = ω m . By using the mode volume V m = 2πL 3 /m(1 − R 1 ) [23], we can express the interaction strength as G m = mG, where we defineG ≡ ( µ ·Ê) c(1 − R 1 )N/4n 3 r ε 0 L 4 . This quantity is a rate that characterises the base collective lightbacteria interaction strength. Instead of fixing the value ofG, we vary this quantityG = [0, 0.1]ω b , which is within experimentally achievable regime (cf. Refs. [11,12]).
Entanglement dynamics (logarithmic negativity) is illustrated in Fig. 2, where we show the existence of steady-state entanglement between the cavity modes. Fig. 2 (a) shows that contribution to 12 : 34 entanglement from the 5th and 6th mode is not significant. Therefore, we consider 4 cavity modes in all other calculations. Apart from bipartition 12 : 34 one can also use other bipartitions as indication of non-classical bacteria, exemplary ones are presented in Fig. 2 (b). In recent experiments, the rateG was shown to be 0.03 ω b [12]. In our calculations we vary this rate, e.g. see Fig. 2 (c) and (d). It is also apparent that entanglement between cavity modes and bacteria E 1234:b grows much faster than entanglement between cavity modes. More precisely, nonzero E 12:34 implies nonzero E 1234:b . Moreover, it can be seen that it takes ∼ 20 fs to reach the steady-state regime, which is much faster than relaxation processes (∼ ps) occuring within green sulphur bacteria [16].
In the steady state regime, we calculated the entanglement in the partition 12 : 34 and 1234 : b, cf. Fig. 3. One can see that the bacteria can be strongly correlated with cavity modes, much stronger than entanglement between the modes. The latter is in the order of 10 −2 − 10 −3 . We note that entanglement in the range 10 −2 has already been observed experimentally between mechanical motion and microwave cavity fields [24].
In order to justify the low atomic excitation limit we plot the evolution of excitation of bacteria (together with the number of photons in different cavity modes) in Fig.  4. It shows that excitation numbers are oscillating in the "steady state". They are caused by interactions between light and bacteria (Rabi-like oscillations). Setting the interactions off (G m = 0) produces constant steady-state values (dashed lines in Fig. 4) given by is the laser-cavity detuning. Note that photon number of the 2 nd cavity mode (solid cyan line) is showing oscillations well bellow its "off-interaction" value (dashed cyan line). This is because ω 2 is in resonance with the frequency of the atomic transition ω b . Finally, we note that the excitation number of the bacterial system can reach ∼ 3000, and with ∼ 10 8 actively coupled dipoles in the cavity [12] it gives ∼ 3 × 10 −3 % excitation, which justifies the low-excitation approximation. Discussion. It should be pointed out that covariance matrix V (t) (and hence the entanglement) does not depend on the power of the lasers. This is a consequence of the dipole-dipole coupling and classical treatment of the driving field (see Appendix B). Hence, the system gets entangled also in the absence of the lasers. There is no fundamental reason why this entanglement with vacuum could not be measured, but practically it is preferable to pump the cavity in order to improve the signal-to-noise ratio. Of course quantities other than entanglement may depend on driving power, for example the light intensity inside the cavity as shown in Fig. 4.
This finding is quite different from results in optomechanical system where the covariance matrix depends on laser power [14,25]. The origin of this discrepancy is the nature of the coupling. For example, in optomechanical system consisting of a single cavity modeâ and a mechanical mirrorb the coupling is proportional toâ †âx b , which is third order in operators [26]. This results in effective coupling strength being proportional to the classical cavity signal α after linearisation of the Langevin equations. This classical signal enters the covariance matrix via effective coupling strength and introduces the dependence on driving power.
We also performed similar calculations in which we neglected the counter rotating terms in Eq. (1), the model known as Tavis-Cummings. This resulted in no entanglement generated in the steady state and can be intuitively understood as follows. Since the steady-state covariance matrix does not depend on initial state, we might start with all atoms in the ground state and vacuum for the light modes. It can be argued in Schrödinger picture that this will be the state of affairs at any time as we only consider energy preserving terms in the light-bacteria coupling. Therefore, nonzero entanglement observed in experiments will provide evidence of the counter rotating terms in the coupling.
Conclusions. We have proposed a setup for the inference of non-classicality of photosynthetic organisms. Our scheme is based on coupling the organisms to a few optical modes in a driven cavity. Our detection method does not require detailed modelling of the organisms and makes no assumptions about interaction between them and the probes. It only requires that the organisms independently interact with each probe for some time after which one estimates entanglement between the probes. Non-classicality is inferred by the observation of nonzero entanglement. We have performed simulations of this system within a plausible model using recent experimental parameters. Calculations confirm measurable steadystate entanglement which implies entanglement with the organisms and provides evidence for strong coupling.
Acknowledgments. This work is supported by the National Research Foundation (Singapore) and Singapore Ministry of Education Academic Research Fund Tier 2 Project No. MOE2015-T2-2-034. CM is supported by the Templeton World Charity Foundation and the Eutopia Foundation. MP is supported by the SFI-DfE Investigator Programme (Grant 15/IA/2864) and a Royal Society Newton Mobility Grant (NI160057).

Appendix A: Evolution of quadratures
The Langevin equations for the quadratures can be written in a simple matrix equationu(t) = Ku(t) + l(t), with the vector u = ( where the components are 2 × 2 matrices given by and 0 is a 2 × 2 zero matrix. We split the last term into two parts, representing the noise and pumping respectively, i.e. l(t) = n(t) + p(t) where (A3) We have also used quadratures for the noise terms, i.e. throughF m = (X m + iŶ m )/ √ 2. The solution to the Langevin equations is given by Eq.
(3) in the main text. This allows numerical calculation for expectation value of the quadratures as a function of time, i.e. u i (t) is given by the i th element of Since every component of p(t) is not an operator, we have p k (t) = tr(p k (t)ρ) = p k (t). Also, we have used the fact that the noises have zero mean, i.e. n k (t) = 0.

Appendix B: Covariance matrix
Covariance matrix fully characterises our system and is defined as . This means that p(t) does not contribute to ∆u i (t) (and hence the covariance matrix) since p k (t) = p k (t). We can then construct the covariance matrix at time t from Eq. (3) without considering p(t) as follows where D = Diag[κ 1 , κ 1 , · · · , κ M , κ M , γ, γ] and we have assumed that the initial quadratures are not correlated with the noise quadratures such that the mean of the cross terms are zero. A more explicit solution of the covariance matrix, after integration in Eq. (B1), is given by which is linear and can be solved numerically. The steady state is guaranteed when all real parts of the eigenvalues of K are negative, i.e. W + (∞) = 0. In this case the covariance matrix satisfies Eq. (4) from the main text.
where N is the total number of modes, which is M + 1 in our case. The block component L jk is a 2 × 2 matrix describing local mode correlation when j = k and intermodal correlation when j = k. An N -mode covariance matrix has symplectic eigenvalues {ν k } N k=1 that can be computed from the spectrum of matrix |iΩ N V | [27] where For a physical covariance matrix 2ν k ≥ 1 [28]. Entanglement is calculated as follows. For example, the calculation in the partition 12 : 34 only requires the following covariance matrix that can be obtained from Eq. (C1). If the covariance matrix is not physical after partial transposition with respect to mode 3 and 4 (this is equivalent to flipping the sign of the operatorŷ 3 andŷ 4 in V 1234 ) then our system is entangled. This unphysical V T34 is shown by its minimum symplectic eigenvaluesν min < 1/2. Entanglement is then quantified by logarithmic negativity as follows E 12:34 = max[0, − ln (2ν min )] [29,30]. Note that the separability condition, whenν min ≥ 1/2, is sufficient and necessary when one considers 1 : N mode partition [31], e.g. 1234 : b.

Appendix D: Excitation numbers
The excitation number of cavity modes and bacteria as a function of time can be calculated from u i (t) and V ii (t). For example, the mean excitation number for the first cavity mode is given by