Thought experiment

Our idea is to design a setup which, on one hand, is close to what has already been realised with bacteria and light, in order to utilise their strong coupling, and whose description, on the other hand, can be phrased within the framework of ref. 26 It was shown there that two physical systems, A and B, coupled via a mediator C, i.e. described by a total Hamiltonian of the form H AC + H BC , can become entangled only if quantum discord D AB|C is generated during the evolution. This also holds if each system is allowed to interact with its own local environment. Therefore, observation of quantum entanglement between A and B is a witness of quantum discord D AB|C during the evolution if one can ensure the following conditions:

(i) A and B do not interact directly, i.e. there is no term H AB in the total Hamiltonian. (ii) All environments are local, i.e. they do not interact with each other. (iii) The initial state is completely unentangled (otherwise entanglement between A and B can grow via classical C26).

We now propose a concrete scheme for revealing non-classicality of the bacteria and argue how it meets these conditions. Consider the arrangement in Fig. 1. The bacteria are inside a driven single-sided multimode Fabry–Perot cavity where they interact independently with a few cavity modes. The cavity modes are divided into two sets which play the role of systems A and B in the general framework. The bacteria are mediating the interaction between the modes and hence they represent system C. Condition (i) above can be realised in practice in at least two ways. An experimenter could utilise the polarisation of electromagnetic waves and group optical modes polarised along one direction to system A and those polarised orthogonally to system B. Another option, which we will study in detail via a concrete model below, is to choose different frequency modes and arbitrarily group them into systems A and B. Condition (ii) holds under typical experimental circumstances where the environment of the cavity modes is outside the cavity whereas that of the bacteria is inside the cavity or even part of bacteria themselves. The electromagnetic environment outside the cavity is a large system giving rise to the decay of cavity modes but having no back-action on them. Therefore, each cavity mode decays independently and cannot get entangled via interactions with the electromagnetic environment. Finally, condition (iii) is satisfied right before placing the bacteria into the cavity, because at this time all three systems A, B, and C are in a completely uncorrelated state ρ A ⊗ ρ B ⊗ ρ C .

Fig. 1 Experimental setup revealing quantum features of photosynthetic organisms. We consider a driven single-sided multimode Fabry–Perot cavity embedding green sulphur bacteria. Here, R 1 is the reflectivity of the input mirror, while the end mirror is perfectly reflecting with R 2 ≈ 1. A few cavity modes individually interact with the bacteria, but not with each other. Both the bacteria and cavity modes are open systems. In particular, the interaction between the bacteria and their environment results in the energy decay rate 2γ n . The mth cavity field mode experiences energy dissipation at a rate 2k m Full size image

We note again that this discussion is generic with almost no modelling of the involved systems. In particular, nothing has been assumed regarding the physics of the bacteria and their interactions with light and the external world. This makes our proposal experimentally attractive. Note also that one can think of the bacteria as a channel between the cavity modes A and B. The method then detects non-classicality of this channel.27,28

In order to make concrete predictions about the amount of intermodal entanglement E A:B we now study a specific model for the energy of the discussed system. This additional assumption about the overall Hamiltonian will allow us to demonstrate that the entanglement E A:B is accompanied by light–bacteria entanglement E AB:C . This independently confirms the presence of light–bacteria discord as entanglement is a stronger form of quantum correlations than discord.7,8,9 In the remainder of the paper we will therefore only calculate entanglement.

Model

We consider a photosynthetic bacterium, Chlorobaculum tepidum, that is able to survive in extreme environments with almost no light.29 Each bacterium, which is ~2 μm × 500 nm in size, contains 200–250 chlorosomes, each having 200,000 bacteriochlorophyll c (BChl c) molecules. Such pigment molecules serve as excitons that can be coupled to light.12,30 The extinction spectrum of the bacteria (BChl c molecules) in water shows two pronounced peaks, at wavelengths λ I = 750 nm and λ II = 460 nm (see Fig. 1b of ref. 12). We therefore model the light-sensitive part of the bacteria by two collections of N two-level atoms with transition frequencies (Ω I , Ω II ) = (2.5, 4.1) × 1015 Hz. Simplification of this model to atoms with a single transition frequency was already shown to be able to explain the results of recent experiments.12,30 This simplification was adequate because only one cavity mode was relevant in the previous experiments. In contrast, several cavity modes are required for the observation of intermodal entanglement and it is correspondingly more accurate to include also all relevant transitions of BChl c molecules. We assume that the molecules (two-level atoms in our model) are coupled through a dipole-like mechanism to each light mode. For N ≫ 1, such collections of two-level systems can be approximated to spin N/2 angular momenta. 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 Holstein–Primakoff transformation.31 This allows us to cast the energy of the overall system as

$$\begin{array}{*{20}{l}} H \hfill & = \hfill & {\mathop {\sum}\limits_m \hbar \omega _m\hat a_m^\dagger \hat a_m + \mathop {\sum}\limits_n \hbar \Omega _n\hat b_n^\dagger \hat b_n} \hfill \\ {} \hfill & {} \hfill & { + \mathop {\sum}\limits_{m,n} \hbar G_{mn}\left( {\hat a_m + \hat a_m^\dagger } \right)\left( {\hat b_n + \hat b_n^\dagger } \right)} \hfill \\ {} \hfill & {} \hfill & { + \mathop {\sum}\limits_m i\hbar E_m\left( {\hat a_m^\dagger {\mathrm {e}}^{ - i\Lambda _mt} - \hat a_m{\mathrm {e}}^{i\Lambda _mt}} \right).} \hfill \end{array}$$ (1)

Here, m = 1,…,M is the label for the mth cavity mode, whose annihilation (creation) operator is denoted by \(\hat a_m\) \(\left( {\hat a_m^\dagger } \right)\) and having frequency ω m . Moreover, both harmonic oscillators describing the bacteria are labelled by n = I,II with \(\hat b_n\) \(\left( {\hat b_n^\dagger } \right)\) denoting the corresponding bosonic annihilation (creation) operator. Each oscillator is coupled to the mth cavity field at a rate G mn . The collective form of the coupling allows us to write \(G_{mn} = g_{mn}\sqrt N\) with \(g_{mn} = \mu _n\sqrt {\omega _m/2\hbar \varepsilon _{\mathrm {r}}\varepsilon _0V_m}\), where μ n is the dipole moment of the nth two-level transition, ε r relative permitivity of medium, and V m the mth mode volume24 (see also refs 32,33 for similar treatments). The cavity is driven by a multimode laser, each mode having frequency Λ m , amplitude \(E_m = \sqrt {2P_m\kappa _m/\hbar \Lambda _m}\), power P m , and amplitude decay rate of the corresponding cavity mode 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 terms \(\hat a_m\hat b_n\) and \(\hat a_m^\dagger \hat b_n^\dagger\). 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 the light-sensitive part of the bacteria to give rise to Markovian open-system dynamics, which is modelled as decay of the two-level systems. For justification we note that in actual experiments the bacteria are surrounded by water which can be treated as a standard heat bath and although the environment of interest cannot be in a thermal state (because the bacteria are alive) its state is expected to be quasi-thermal. Given that the bacterial environment of the BChl c molecules is of finite size we should also justify the Markovianity assumption. To the best of our knowledge there is no experimental evidence against this assumption. Likely this is due to the fact that all excitations arriving at this environment are further rapidly dissipated to the large thermal environment of water, whose energy is small compared to the optical transitions.

We treat the environment of the cavity modes as the usual electromagnetic environment outside the cavity.34,35 This results in independent decay rates of each mode. Taken all together, the dynamics of the 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 environments

$$\begin{array}{*{20}{l}} \dot{{\hat{a}}}_{m} \hfill & = \hfill & { - (\kappa _m + i\omega _m)\hat a_m - i\mathop {\sum}\limits_n G_{mn}\left( {\hat b_n + \hat b_n^\dagger } \right) + E_m{\mathrm {e}}^{ - i\Lambda _mt}} \hfill \\ {} \hfill & {} \hfill & { + \sqrt {2\kappa _m} \hat F_m,} \hfill \\ \dot{{\hat{b}}}_{m} \hfill & = \hfill & { - \left( {\gamma _n + i{\mathrm{\Omega }}_n} \right)\hat b_n - i\mathop {\sum}\limits_m G_{mn}\left( {\hat a_m + \hat a_m^\dagger } \right) + \sqrt {2\gamma _n} \,\hat Q_n,} \hfill \end{array}$$ (2)

where γ n is the amplitude decay rate of the bacterial system. \(\hat F_m\) and \(\hat Q_n\) are operators describing independent zero-mean Gaussian noise affecting the mth cavity field and the nth bacterial mode, respectively. The only nonzero correlation functions between these noises are \(\langle \hat F_m(t)\hat F_{m^{\prime}}^\dagger (t^\prime )\rangle = \delta _{mm^\prime }\delta (t - t^\prime )\) and \(\langle \hat Q_n(t)\hat Q_{n^\prime }^\dagger (t^\prime )\rangle = \delta _{nn^\prime }\delta (t - t^\prime )\).34,35 We note that in this model the light-sensitive part of the bacteria is treated collectively, i.e. all its two-level atoms are indistinguishable. This assumption is standardly made in present-day literature, see e.g. refs 12,30 where modelling of the bacteria/chlorosomes as a harmonic oscillator fits observed experimental results. But it should be stressed that this assumption deserves an in-depth experimental assessment.

We express the Langevin equations in terms of mode quadratures. In particular, by using \(\hat x_m \equiv (\hat a_m + \hat a_m^\dagger )/\sqrt 2\) and \(\hat y_m \equiv (\hat a_m - \hat a_m^\dagger )/i\sqrt 2\) one gets a set of Langevin equations for the quadratures that can be written in a matrix equation \(\dot u(t) = Ku(t) + l(t)\) with the vector \(u = (\hat x_1,\hat y_1, \cdots ,\hat x_M,\hat y_M,\hat x_{\mathrm{I}},\hat y_{\mathrm{I}},\hat x_{{\mathrm{II}}},\hat y_{{\mathrm{II}}})^T\). Here, K is a square matrix with dimension 2(M + 2) describing the drift and l is a 2(M + 2) vector containing the noise and pumping terms (see the Methods section for explicit expressions). The solution to the Langevin equations is given by

$$u(t) = W_ + (t)u(0) + W_ + (t){\int}_0^t {\mathrm {d}}t^\prime W_ - (t^{\prime} )l(t^{\prime} ),$$ (3)

where W ± (t) = exp(±Kt).

One can construct the covariance matrix as a function of time V(t) from Eq. (3) (cf. Methods section). Time evolution of important quantities can then be calculated from the covariance matrix, e.g. entanglement and excitation number (cf. Methods section). 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

$$K\,V(\infty ) + V(\infty )\:K^T + D = 0,$$ (4)

where D = Diag[κ 1 , κ 1 ,…,κ M , κ M , γ I , γ I , γ II , γ II ]. Note that the steady-state covariance matrix does not depend on the initial conditions, i.e. V(0). Moreover, as the Langevin equations are linear and due to the gaussian nature of the quantum noises, the dynamics of the system is preserving gaussianity. Therefore the steady state is a continuous variable gaussian state completely characterised by V(∞).

Results of calculations

We now calculate the steady-state entanglement using, wherever possible, parameters from the experiments of ref. 12 We place the bacteria in a single-sided Fabry–Perot cavity of length L = 518 nm (cf. Fig. 1). The refractive index due to aqueous bacterial solution embedded in the cavity is \(n_{\mathrm {r}} = \sqrt {\varepsilon _{\mathrm {r}}} \approx 1.33\), which gives the frequency of the mth cavity mode ω m = mπc/n r L ≈ 1.37 m × 1015 Hz. The reflectivities of the mirrors are engineered such that R 2 = 100% and R 1 = 50%. We assume the reflectivities are the same for all the optical modes, giving κ m ≈ 7.5 × 1013 Hz through the finesse \({\cal F} = - 2\pi /{\mathrm{ln}}\left( {R_1R_2} \right) = \pi c/2\kappa _mn_{\mathrm {r}}L\). The decay rate of the excitons can be calculated as γ n = 1/2τ n , where τ n = 2h/Γ n is the coherence time with Γ n being the full-width at half-maximum (FWHM) of the bacterial spectrum.36 We approximate the spectrum in Fig. 1b of ref. 12 as a sum of two Lorentzian functions centred at Ω I and Ω II having FWHM of (Γ I , Γ II ) = (130, 600) meV, giving (γ I , γ II ) ≈ (0.78, 3.63) × 1013 Hz, respectively. Note that the decay rate solely depends on the coherence time, i.e. we assume only homogenous broadening of the spectral lines.

All the spectral components of the driving laser are assumed to have the same power P m = 50 mW and frequency Λ m = ω m . By using the mode volume V m = 2πL3/m(1 − R 1 ),37 we can express the interaction strength as \(G_{mn} = m\tilde G_n\), where we define \(\tilde G_n \equiv \mu _n\sqrt {c(1 - R_1)N/4\hbar n_{\mathrm {r}}^3\varepsilon _0L^4}\). This quantity is a rate that characterises the base collective interaction strength of the cavity mode and the nth bacterial mode. Instead of fixing the value of \(\tilde G_n\), we vary this quantity \(\tilde G_n = [0,0.2]10^{15}\,{\mathrm{Hz}}\), which is within experimentally achievable regime (cf. refs 11,12).

Logarithmic negativity is chosen as entanglement quantifier and the Methods section provides the details on how this quantity is calculated. We consider four cavity modes as the addition of higher modes shows negligible effects to the steady-state entanglement. In the steady-state regime, we calculate entanglement between the cavity modes E 12:34 , between the cavity modes and bacteria E 1234:I II , and between the bacterial modes E I:II , cf. Figure 2. This steady-state regime is reached in ~100 fs (see Methods), which is faster than relaxation processes (~ps) occuring within green sulphur bacteria.30 Our results show that the steady-state entanglement E 12:34 is always accompanied by E 1234:I II , i.e. the bacteria are non-classically correlated with the cavity modes. This is in agreement with the general detection method of ref. 26 as entanglement is a stronger type of quantum correlation than discord, i.e. nonzero E 1234:I II implies nonzero cavity modes-bacteria discord D 1234|I II . Our results also show that the entanglement dynamics of E 12:34 is dominated by modes 2 and 3 since other modes are further off resonance with the bacterial modes. Moreover, there is entanglement generated within the bacteria. This requires both \(\tilde G_{\mathrm{I}}\) and \(\tilde G_{{\mathrm{II}}}\) to be nonzero and relatively high. We see that the bacteria can be strongly entangled with the cavity modes, much stronger than entanglement between the cavity modes. While 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.38 We have also indicated, as black dots in Fig. 2, the coupling strengths \(\tilde G_{\mathrm {I}} = 3.9 \times 10^{13}\,{\mathrm{Hz}}\) from ref. 12 and the corresponding \(\tilde G_{{\mathrm{II}}} = 6 \times 10^{13}\,{\mathrm{Hz}}\), which is estimated as follows. From the relation \(\mu _n^2 \propto {\int} f(\omega ){\mathrm {d}}\omega /\omega _n\),39 where f is the extinction coefficient, one can obtain the ratio \(\tilde G_{{\mathrm{II}}}/\tilde G_{\mathrm{I}} = \mu _{{\mathrm{II}}}/\mu _{\mathrm{I}} \approx 1.53\).