# Holographic Photosynthesis

###### Abstract

There are successful applications of the holographic AdS/CFT correspondence to high energy and condensed matter physics. We apply the holographic approach to photosynthesis that is an important example of nontrivial quantum phenomena relevant for life which is being studied in the emerging field of quantum biology. Light harvesting complexes of photosynthetic organisms are many-body quantum systems, in which quantum coherence has recently been experimentally shown to survive for relatively long time scales even at the physiological temperature despite the decohering effects of their environments.

We use the holographic approach to evaluate the time dependence of entanglement entropy and quantum mutual information in the Fenna-Matthews-Olson (FMO) protein-pigment complex in green sulfur bacteria during the transfer of an excitation from a chlorosome antenna to a reaction center. It is demonstrated that the time evolution of the mutual information simulating the Lindblad master equation in some cases can be obtained by means of a dual gravity describing black hole formation in the AdS-Vaidya spacetime. The wake up and scrambling times for various partitions of the FMO complex are discussed.

###### Keywords:

holography, AdS/CFT correspondence, photosynthesis, light-harvesting complex, black holes, entanglement entropy, mutual information^{†}

^{†}institutetext: Steklov Mathematical Institute, Russian Academy of Sciences,

Gubkina str. 8, 119991, Moscow, Russia

## 1 Introduction

The anti-de Sitter - conformal field theory (AdS/CFT) correspondence Aharony:1999ti and more general holographic gravity/gauge duality play an important role in modern theoretical physics. They have been used for description of strong interacting equilibrium and non-equilibrium systems in high energy physics, in particular, to describe heavy-ion collisions and formation of quark-gluon plasma Solana ; IA ; DeWolf , as well as in condensed matter physics Hartnoll08kx ; Sachdev:2010ch .

According to the holographic correspondence the quantum gravity (string theory) on anti-de Sitter spacetime (AdS) is equivalent to a certain quantum field theory on the AdS boundary. The holographic approach provides a powerful method for studying strongly coupled quantum field theories by means of the dual classical gravitational theory in the AdS space which is more tractable.

In last years, there has been a growing interest to study the entanglement entropy and quantum mutual information for various quantum systems by using the holographic approach Ryu:2006bv -Mirabi:2016elb and refs therein. The entanglement entropy of a boundary region is determined by the area of the minimal bulk surface that coincides with the entangling surface at the boundary Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt .

In this paper we apply the holographic approach to photosynthesis. Light harvesting complexes (LHC) in bacteria and plants are important examples of nontrivial quantum phenomena relevant for life which are being studied in the emerging field of quantum biology. Recent investigations of quantum effects in biology include also the process of vision, the olfactory sense, the magnetic orientation of migrant birds as well as photon antibunching in proteins, the quantum delocalization of biodyes in matter-wave interferometry and quantum tunneling in biomolecules, see for instance Markus Arndt ; Graham R. Fleming ; OhyaVol ; QBIC ; Quantum Effects in Biology ; IV .

Photosynthesis is vital for life on Earth Blanck . Photosynthesis changes the energy from the sun into chemical energy and splits water to liberate oxygen and convert carbon dioxide into organic compounds, especially sugars. Nearly all life either depends on it directly as a source of energy, or indirectly as the ultimate source of the energy in their food.

Light-harvesting complexes in plants and photosynthetic bacteria include protein scaffolds into which pigment molecules are embedded, e.g. chlorophyll or bacteriochlorophyll molecules. The pigment molecules absorb light and the resulting electronic excitation, exciton, is transported between the pigment molecules until it reaches a reaction center complex, where its energy is converted into separated charges. The process whereby the light energy is transported through the cell is extremely efficient – higher than any artificial energy transport process.

One models many photosynthetic light harvesting complexes by a general three-part structure comprising the antenna, the transfer network, and the reaction center. The antenna captures photons from sunlight and subsequently excites the electrons of the pigment from their ground state. The excited electrons, which combine with holes to make excitons, travel from the antenna through an intermediate protein exciton transfer complex to the reaction center where they participate in the chemical reaction that generates oxygen.

Quantum coherences have been observed in two-dimensional spectroscopic studies of energy transfer within several different light harvesting complexes. The simplest light harvesting complex is the Fenna-Matthews-Olson (FMO) protein-pigment complex in green sulfur bacteria. We demonstrate that some numerical results on the time evolution of the mutual information for the FMO complex FM can be obtained by using the holographic approach.

Experiments with the FMO complex have shown the presence of quantum beats between excitonic levels at both cryogenic and ambient temperatures Engel ; Collini ; Panit ; SFOG . Quantum coherence has also been seen in light harvesting antenna complexes of green plants Calh and marine algae Coll . There are also studies of quantum coherence within the reaction center Lee .

Theoretical studies of excitation dynamics in the FMO complex in the single excitation subspace have demonstrated the presence of long-lived, multipartite entanglement. A study of the temporal duration of entanglement in the FMO complex using a simulation of excitation energy transfer dynamics under conditions that approximate the real environment was performed in MRL -TAD . Entanglement within the Markovian description of the FMO complex has been considered in Car ; Car-2 . The time evolution of the mutual information in the FMO complex was studied in 0912.5112 by using simulation of the Gorini-Kossakowski-Sudarshan-Lindblad master equation. Quantum nonlocality as a function of time is studied in Charlotta Bengtson .

The paper is organized as follows.

In Sect.2 has an introductory character. Here we remind the main objects and tools of our study of the FMO complex. In Sect.2.1 has an introductory character. Here we briefly describe the FMO complex and write down the corresponding master equation. In Sect.2.1.3 the standard consideration of LHC as quantum system and entropy of entanglement and mutual information for the FMO complex are sketched. In Sect.2.1.4 we list different reductions of the FMO complex used in modern theoretical studies. In Sect.2.2 definitions of holographic entanglement entropy and mutual information, as well as the basic formula that we use in the main text are presented. In Sect.2.3 two iterative procedures to calculate the holographic entanglement entropy are presented. The fist one takes into account contributions only of primitive diagrams and the second one incorporates also Boltzmann rainbow diagrams. In Sect.2.4 we remind the phase structure of the holographic mutual information for two belts in the static backgrounds, empty AdS and AdS black brane.

Sect.3 is devoted to the study of the time evolution of holographic mutual information for the simplest reduction of the FMO complex. We start, Sect.3.1, by calculation of holographic entanglement entropy for two site system during a quench at nonzero temperature. Then in Sect.3.2 we study holographic mutual information for two site system at nonzero temperature. In Sect.3.3 we compare the results of our calculations with the mutual information calculated for the reduced FMO system in 0912.5112 . In Sect.3.4 we discuss the dependencies of wake up time and scrambling times on the geometrical parameters and the initial temperature.

In Sect.4 we consider the holographic mutual information for the reduction of the FMO complex with one composite part. For this purpose in Sect.4.1 we study the phase structure of holographic entanglement entropy for the Vaidya shell in the AdS black brane background. In Sect.4.2 the mutual information for 3 segments and scrambling time for this background is calculated and in Sect.4.3 we make a fit of time dependence of the mutual information at the physiological temperature () calculated in 0912.5112 for one mixed state by the time dependence of the holographic mutual information under the global quench by the Vaidya shell in AdS.

## 2 Setup

### 2.1 FMO complex

#### 2.1.1 Seven bacteriochlorophills

The FMO protein complex FM is the main light-harvesting component of the green sulfur bacteria Prosthecochloris
aestuarii. It is a trimer,
consisting of three identical molecular sub-units, Fig.1A.
Each of the sub-units is a network
of seven^{1}^{1}1Recently, an additional bacteriochlorophyll (the eighth) pigment was discovered in each subunit of this trimerSchmidt . interconnected bacteriochlorophylls
arranged in two weakly connected
branches that are separately connected to the antenna (bacteriochlorophyll sites one and six) and jointly connected to the reaction center via site three, Fig.1B.

The theoretical models in the literature Car-2 ; Sar ; 0912.5112 ; Car study the dynamics of one sub-unit of the trimer. The total Hamiltonian of the system includes the non-relativistic QED Hamiltonian in the dipole approximation, phonon Hamiltonian and other environmental fields, for a derivation of the master equation in the weak coupling stochastic limit see ALV and also AKV ; KV .

We consider one sub-unit of the trimer which consists of seven bacteriochlorophyll sites transferring energetic excitations from a photon-receiving antenna to a reaction center, see Fig.1.

A

#### 2.1.2 Master equation

We consider the nine-state model Sar ; 0912.5112 for excitation transfer in the single excitation approximation which is described by a 9-dimensional subspace in the Hilbert space . The possible states for the exciton will be expressed in the site basis where the state indicates that the excitation is present at site . There are also a ground state corresponding to the loss or recombination of the excitation and a sink state corresponding to the trapping of the exciton at the reaction center (there is a discussion of the ”local” and ”global” bases in theory of open quantum systems in TV ). The density operator for this quantum system has the following representation in the site basis:

(2.1) |

It is assumed that the density matrix satisfies the GKS-Lindblad master equation of the following form

(2.2) |

Here the Hamiltonian describes the coupling between the seven sites states :

(2.3) |

where is the energy of the site and describes the coupling between sites an . A Lindblad superoperator has the general form

(2.4) |

where are arbitrary operators and are positive constants, see for example OhyaVol . In our case for the FMO complex the Lindblad superoperator is taken in the form MRL ; Car-2 ; Sar ; 0912.5112

(2.5) |

Here the first term describes the dissipative recombination of the exciton,

(2.6) |

where is the rate of the recombination at site .

The second Lindblad superoperator accounts for the dephasing interaction with the environment,

(2.7) |

where is the rate of dephasing at site .

The final term accounts for the trapping of the exciton in the reaction center:

(2.8) |

It is supposed that the initial state of the FMO complex is a pure excitation at site one or site six, or a mixture of these two states.

#### 2.1.3 Entropy of entanglement and mutual information

Let two parties and share a quantum state in a Hilbert space . The von Neumann entropy of this state is

(2.9) |

Similarly, one can define the entanglement entropies

(2.10) |

where the reduced density matrices and .

The quantum mutual information measures the correlations shared between the two parties,

(2.11) |

This can be written as a relative entropy and is therefore non-negative:

(2.12) |

where

see for example OhyaVol .

Several simulations of the quantum mutual information as a function of time at cryogenic and physiological temperatures were conducted in 0912.5112 . In particular the simulations calculate the quantum mutual information with respect to several ”bipartite cuts” of the sites in the FMO complex. The different cases are as follows, (Figs.1-6 in 0912.5112 ):

1. The first cut was picked up where system consists of site three and system consists of sites one and six, (Figs.1 and 2 in 0912.5112 ). Remind that the initial state of the complex is at site one, six, or the mixture, and the FMO complex transfers the excitation from these initial sites to site three.

2. The next bipartite cut is with the A system consisting of sites one and two and the B system consisting of site three, (Figs.3 and 4 in 0912.5112 ).

3. Finally, the third cut was the cut where system A consists of site three and system B consists of all other sites, (Figs.5 and 6 in 0912.5112 ). The quantum mutual information for this case should be larger than for the case of the other cuts.

#### 2.1.4 Reductions of the FMO complex

We start from Fig.1. Several reductions of this system to more simple ones are considered. Some of them are schematically presented in Fig.2–Fig.4.

### 2.2 Holographic entanglement entropy

#### 2.2.1 Static AdS background

Consider a quantum field theory on a -dimensional manifold , where and denote the time axis and the -dimensional space-like manifold, respectively. Let be given a -dimensional submanifold at fixed time and let be its boundary. Then the formula for the holographic entanglement entropy in a CFT on reads Ryu:2006bv

(2.13) |

where is the area of , that is the dimensional static minimal surface in AdS with metric

(2.14) |

whose boundary is given by , and is the dimensional Newton constant, is the radius of AdS.

The simplest example for the shape of is a straight belt at the boundary , see Fig.5,

(2.15) |

In this case the area of the minimal surface, divided on the suitable constants, is Ryu:2006bv

(2.16) |

where is the length of in the traversal -direction, is the UV regularization, is a positive constant depending on . The first term is divergent when , but for our purpose it will not play a role, since we will consider the differences of entanglement entropies.

For an excited state whose gravitational dual is provided by the black brane solution with mass and the Hawking temperature (here we assume )

(2.17) |

in general, it is not possible to find an explicit expression for the entanglement entropy for . It can be found from the integral equation relating the entropy to an auxiliary parameter

meanwhile the parameter is related to the width of the belt

#### 2.2.2 Time dependent AdS Vaidya background

The proposal (2.13) has been generalized to time dependent geometries in Hubeny:2007re . In the Vaidya AdS spacetime the corresponding minimal surface describes the thermalization process in the d-dimensional boundary theory Balasubramanian:2011ur , see also AbajoArrastia:2010yt ; Albash:2010mv ; Allais:2011ys . The Vaidya AdS metric has the form

(2.19) |

where the function

(2.20) |

The form of is usually chosen to be

(2.21) |

The metric (2.19) with this function describes a spacetime model which evolves from pure AdS at early times to the Schwarzschild black brane at late times because of the shell of null dust infalling along . The parameter determines the thickness of the shell. The case corresponds to a step function and one deals with a shock wave.

We will consider also the shell in the AdS black brane background (the Vaidya AdS black brane metric), in this case

(2.22) |

The entanglement entropy is given by the extremum of the functional Balasubramanian:2011ur ; Allais:2011ys

(2.23) |

where and .

The Euler-Lagrange equations corresponding to the action (2.23) read

(2.24) | |||||

(2.25) |

For the system of equations (2.24), (2.25) we will solve numerically the Cauchy problem with initial data (2.27), (2.27)

(2.26) |

(2.27) |

We are interested in finding solutions that reach the boundary at some point, that we identify^{2}^{2}2In our calculation the width of the belt is , although on some pictures we omit the factor . with in (2.23) at the boundary time :

(2.28) |

Equations (2.24), (2.25) have an integral of motion

(2.29) |

Due to this identity the integral in (2.23) can be substantially simplified to give

(2.30) |

where we introduce regularization . This integral contains the UV divergence when . The renormalized version of (2.30) can be written as Balasubramanian:2011ur ; Allais:2011ys

(2.31) |

and regularization can be removed. Note, that in the case of Balasubramanian:2011ur ; Allais:2011ys

(2.32) |

It is quite straightforward to find the dependence of the renormalized entanglement entropy on and . To find the dependence of the entanglement entropy on and one has to parametrize the solutions to (2.27)-(2.25), not by , but by and .

It happens that the values of and are very sensitive to the initial data (2.26), and one needs a special numerical procedure to find the pair corresponding to the given pair with large values of , see Allais:2011ys ; 1512.05666 ; 1601.06046 and refs therein for more details. It is interesting to note that the case when in (2.22) is more stable than the case .

In Fig.7 the dependence of the renormalized holographic entanglement entropy on and for the propagating Vaidya shell in the four dimensional black brane background with given by (2.20) and (2.22) is presented. As compare with the dependence of the holographic entanglement entropy for the Vaidya AdS metric the entropy does not go to fixed value for large , but as both go to constant values for large times. The same is shown in Fig.8 for the three dimensional black brane background.

Note that the similar technique has been intensively used in study the thermalization processes, especially for non-conformal invariant backgrounds, see Alishahiha:2014cwa ; 1401.6088 ; 1503.02185 ; 1512.05666 ; 1601.06046 and refs therein.

### 2.3 Iterative procedure to calculate

In this section we present the rules that help to calculate the holographic entropy for n disjoint objects. Here we consider for illustration a particular case of -segments. This problem has been studied in several papers Balasubramanian:2011at ; Hayden:2011ag ; Allais:2011ys ; Alishahiha:2014jxa ; Ben-Ami:2014gsa ; Mirabi:2016elb . It is substantially simplified in the case of equal length strips and equal separation between them. However in order to deal with holographic description of the FMO complex, we have to find the entropy for non-equal length strips.

To find the entropy one has to find the global minimum among all possible configurations. To specify all possible configurations corresponding to local minimum surfaces it is convenient to use the diagrammatic language. In special cases, it happens that only primitive diagrams contribute to the entropy. The primitive diagrams are diagrams that do not contain cross-sections of the connected lines and also do not contain diagrams with ”engulfed” sub-diagrams in the terminology used in Ben-Ami:2014gsa . Note that the class of primitive diagrams contains less diagram then so-called rainbow diagrams in the Boltzmann quantum field theory IAIV . It is possible that for more complicated backgrounds one has to take into account more diagrams then only the primitive diagrams.