enhanced excitation energy transferin the photosynthesis

C. Negulescu, march 2021

1

Enhanced Excitation Energy Transfer in the

photosynthesis process

Claudia Negulescu

Institut de Mathématiques de Toulouse

Université Paul Sabatier


2Aim and motivation of this work

• Thematic:

Introduction and study of mathematical models for the description of

open quantum systems, embedded in an environment.

• Motivation:

➠ biological processes present capabilities which are very

impressive (activity in the brain, photosynthesis process, etc);

➠ these performances cannot be adequately explained via traditional,

classical approaches;

➠ a certain amount of quantum coherence is thought to be used by

Nature to enhance the underlying processes.

• Questions:

➠ How can quantum features survive in open quantum systems?

➠ Can the same procedures be used for technological applications?


3Signal transfer in neural networks

Theme: Modelling of the exciton transfer in the nervous system

Basic ingredients:

➠ the processing element (neuron, excited/firing or not)

➠ the inter-connection structures between neurons

➠ the network dynamics

➠ the learning rules, governing the inter-connection couplings


4Excitation energy transfer in photosynthesis

Theme: Modelling of the coherent exciton energy transfer in

biological systems

Photosynthesis:

➠ process by which plants transform light energy into chemical one

➠ leafs (chlorophyll molecules) capture a wide spectrum of sun’s

energy, transfer the absorbed photons (excitation transfer) towards

a reaction center, where the reaction takes place

6CO2 + 6H2O →photons 6O2 + C6H12O6

➠ carbon dioxide consumed, oxygen released, carbohydrate stored!


5Performance of photosynthesis process

Biological processes present impressive capabilities and performances,

which are still not well understood with traditional approaches.

Exciton energy transfer from pigment to pigment:

➠ what is the mechanism behind the performance of the exciton

energy transfer towards the reaction center?

➠ is it an uncoherent hopping (classical model), a coherent energy

transfer (quantum mechanical mechanism) or an intermediate

regime (environmental assissted transport)?

➠ if the mechanism involves quantum

mechanical means, how can it be,

as photosynthesis takes place in a wet,

warm, noisy environment ⇒

decoherence?


6Quantum signature

• Distinctive characteristics of Quantum mechanics:

➠ discreteness, wave-particle dualism, tunneling effect, coherence, ...

➠ coherence is one of the most striking illustrations of QM

• Quantum mechanics allows for superposition states:

➠ normalized sums of admissible wave-functions are once more

admissible states

➠ possibility to construct non-localized states, which have no

classical counterpart

➠ observable mark of such superpositions : interference pattern!

• Disapearance of interference pattern at human scale!

➠ understanding this decoherence phenomenon is crucial!


7Open quantum systems

• Linear Schrödinger equation describes isolated systems:

i ~ ∂tψ = H ψ , H: Hamiltonian

• Systems in nature are never completely isolated!

➠ central quantum system interacts with its environment, giving rise

to entangled states (which cannot be separated)

• Entanglement is the key concept in the decoherence understanding

➠ entangled states encapsulate correlations btw subspaces

• Density matrix formalism better adapted for entangled systems:

i ~ ∂tρ = [H , ρ] , ρ(t, x, x′) = ψ(t, x)ψ∗(t, x′)

• Tracing over the environ. permits to describe soley the central syst.

ρS = TrE ρ ⇒ reduced density matrix formalism

➠ emergence of classical mechanics, decoherence phenomenon,

expressing the missed correlations by performing the averaging.


8Simple decoherence models

• Overwhelming complexity of biological systems ⇒

simplifications are required!

• Canonical models can be introduced:

➠ Central system S:

→ particle with continuous coord. in phase-space (x,p) as photons

→ discrete two-level system (TLS, spin-1/2 particle)

➠ Environment E :

→ collection of continuous harmonic oscillators (vibrational env.)

→ collection of discrete TLS (spins)

• Choice of the interaction between central syst. and env. is crucial!

• Dynamics of the environment is not so meaningful

→ Master equations allow for the computation of the evolution of the

central syst. only, averaging over the env. influence


9Different mathematical descriptions

Several levels of description are possible for open quantum systems:

• Fully quantum mechanical description Hfq = HS +HE +HI

i ~ ∂tψ = Hfq ψ ⇒ i~ ∂tρqua = L ρqua

• Quantum-classical description

i~ ∂tρqc = [Hqc, ρqc] , Hqc := HS +Hpert(t)

• Classical approach

∂tρcl = {Hcl, ρcl} , {f, g} := ∂xf ∂pg − ∂pf ∂xg

→ these approaches differ in the manner the environ. is modelled, precision

and complexity (dynamics/influence of the env. difficult to capture);

→ the distinction between quantum and classical features is more subtle

than can be thought.


10

The photosynthesis process

First mathematical model


11Spin-Boson model

i~ ∂tΨ(t, ·) = HΨ(t, ·) , (Schrödinger eq.)

• Hamiltonian consists of three parts H = HS +HE +HI

HS :=~

2

N∑

k=1

ωkσzk +

N−1∑

k=1

λk(

σ+k σ−k+1 + σ−k σ

+k+1

)

, (spin-chain)

HE := ~ωc

(

a†a+

1

2Id

)

, (common harm. oscil.)

HI :=~

2

(

a† + a

)

N∑

k=1

gk σzk , (chain-env. dephasing interaction)

a :=1√

2mωc ~(mωc x+ ip) , a

† :=1√

2mωc ~(mωc x− ip)

σ+k σ−

j := Πk−1i=1 Id⊗ σ+ ⊗Πj−1

i=k+1Id⊗ σ− ⊗ΠNi=j+1Id , σ± :=

σx ± iσy

2


12Mathematical model

• Nbr. of excitations is conserved, as [HI , σz] = 0

Ψ ∈ (L2(R;C))N , Ψ(t, ·) := (ψk(t, ·))Nk=1 , ψk(t, ·) = ψ−−···−+−···−

• Hamiltonian restricted to single-excitation space:

H :=

Hosc + ǫ1 + γ1(x) λ1 Id 0

λ1 Id Hosc + ǫ2 + γ2(x) λ2 Id 0

.

.

.. . .

.

.

.

0 0 λN−1 Id Hosc + ǫN + γN (x)

Hosc := − ~2

2m∂xx+

mω2c

2x2 , ǫk :=

~

2

N∑

j=1

s(k)j ωj , γk(x) := x

√

mωc ~

2

N∑

j=1

s(k)j gj

• Degree of freedom of env.: x; degree of freedom of spin-chain: N


13Isolated spin-chain

• Time ev. of occupation proba. for diff. config. for N = 3 and N = 14; λk = λ0 = 20.

0 0.02 0.04 0.06 0.08 0.1

t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

||j(t

)||

L2

2

Occupation probability ||j(t)||

L2

2 , Spin-nbr.:3

j=1

j=2

j=3

0 0.02 0.04 0.06 0.08 0.1

t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

||j(t

)||

L2

2


L2

2 , Spin-nbr.:14

j=1

j=2

j=3

j=4

j=13

j=14

• Excitation arrival time t⋆(N) at the end of the chain; occupation proba. of last. site at t⋆(N)

5 10 15 20 25

N

0.01

0.02

0.03

0.04

0.05

0.06

0.07

t *(N

)

Time of the arrival at final state

5 10 15 20 25

N

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

||N

(t*)|

|L

22

Maximum occupation probability at final state


14Isolated spin-chain

• Time ev. of occupation proba. for diff. config., for N = 20 and

well-chosen coupling strengths λk

0 0.02 0.04 0.06 0.08 0.1

t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

||j(t

)||

L2

2


L2

2 , Spin-nbr.:20

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

||j(t

)||

L2

2


L2

2 , Spin-nbr.:20

j=1

j=2

j=3

j=4

j=19

j=20

1. Case λk := λ0√

k(N − k) with λ0 = 20 ⇒ λmax = λ0N/2

2. Case λk := λ0√

k(N − k) with λ0 = 2λmax/N , λmax fixed.• Remarks:

➠ perfect excitation transfer can be achieved in isolated chains with well modulated coupling

strengths;

➠ excitation arrival time at the end of the chain has to be knwon to extract the excitation;

➠ what is the influence of an environment on this (perfect) excitation energy transfer?


15Non-isolated spin-chain

• Time ev. of occupation proba. for diff. config., for N = 14 and modulated λk + introduction of

the vibrational environment (harmonic oscillator)

0 0.02 0.04 0.06 0.08 0.1

t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

||j(t

)||

L2

2


L2

2 , Spin-nbr.:14

0 0.02 0.04 0.06 0.08 0.1

t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

||j(t

)||

L2

2

Perturbed occupation proba ||j(t)||

L2

2 , Spin-nbr.:14

j=1

j=2

j=3

j=4

j=13

j=14

• Time ev. of the Von-Neumann entropy S(t) := −Tr [ρS(t) ln(ρS(t))]: cst. + modulated λk

0 0.02 0.04 0.06 0.08 0.1

t

0

0.1

0.2

0.3

0.4

0.5

0.6

S(t

)

Entanglement: Entropy evolution

N=3

N=5

N=14

N=20

0 0.02 0.04 0.06 0.08 0.1

t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

S(t

)

Entropy evolution S(t)Spin-nbr.:3

N=20

N=14

N=5

N=3


16Conclusions first model

Done:

➠ introduction of a mathematical model for fully coherent propagation of an excitation

through a spin-chain;

➠ first numerical tests, to understand how to perform a perfect excitation transfer.

Observations:

➠ perfect transfer is achieved for a specific class of Hamiltonians, with well-engineered

coupling strengths among the TLSs;

➠ the influence of the environment on the well-tailored spin-configuration can be rather

drastic;

➠ in the case of a completely isolated system, the quantum spin-chain model is shown to be

equivalent to a classical harmonic oscillator problem

i~ ∂tΨ = HS Ψ ↔ X′′

(t) = −K X(t)


17

The photosynthesis process

Second mathematical model


18Mathematical model

i~ ∂tΨ(t, ·) = HΨ(t, ·) , (Schrödinger eq.)

• Hamiltonian consists of a perturbed central part H = HS

HS :=~

2

N∑

k=1

ωkσzk +

N−1∑

k=1

λk(t)(

σ+k σ−k+1 + σ−k σ

+k+1

)

, (spin-chain)

• The vibrational motion of underlying molecular structure

introduces time-dependent coupling-strengths λl(t), as:

dl(t) := d0 − [zl(t)− zl+1(t)] , zl(t) : displacement wrt. eq.

λl(t) :=λ̃l

[dl(t)/d0]3

dl(t) := d0 [1− 2al sin(ωv t+ ϕl)] , dl(t) := d0

[

1− a e−[(l−1)d0−vt]2

2σ2

]


19Driven two spin chain

• Solution of the Schrödinger equation i~ ∂tΨ(t, ·) = HSΨ(t, ·)

HS(t) := diag(ε1, ε2, · · · , εN )+diag(−1; λ1(t), · · · , λN−1(t))+diag(+1; λ1(t), · · · , λN−1(t))

ǫl :=~

2

N∑

j=1

s(l)j ωj

Ψ(t) = ΣNl=1αl(t)e

−iǫlt/~el , H0el = ǫl el

α′(t) = − i

~

0 λ1 (t)e−i(ǫ2−ǫ1)t/~ 0

λ1(t)ei(ǫ2−ǫ1)t/~ 0 λ2(t)e−i(ǫ3−ǫ2)t/~ 0

.

.

.. . .

.

.

.

0 0 λN−1(t)ei(ǫN−ǫN−1)t/~ 0

α(t) .

• For N = 2, one gets the occupation probabilities Pl(t) := |αl(t)|2 (ξ0 := ǫ2−ǫ1~

) :

|α1(t)|2 = cos2(

1

~

∫ t

0λ(t′) cos(ξ0t

′) dt′)

, |α2(t)|2 = sin2(

1

~

∫ t

0λ(t′) cos(ξ0t

′) dt′)


20Driven two spin chain (~ = 1, ξ0 = 0)

d(t) := d0 [1− 2a sin(ωv t+ ϕ)] , λ(t) :=λ̃

[d(t)/d0]3, a = 1/4 , d0 = λ̃ = 1 , ϕ = π/2

• Occupation probabilities for cst. λavg := 1T

∫ T0 λ(t) dt with T = 2π

ωvand ωv = ω⋆ = 4.5

0 1 2 3 4 5 6 7t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P1

,2(t

)

Site occupation probabilities, avg

• Time evolution of λ(t) for ωv = ω⋆ = 4.5 and corresp. P1,2(t)

0 1 2 3 4 5 6 7t

0

1

2

3

4

5

6

7

8

(t)

Time-dependent coupling strength

avg=2.3

min=0.3

0 1 2 3 4 5 6 7t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P1

,2(t

)

Site occupation probabilities, *(t)


21Driven two spin chain (~ = 1, ξ0 = 0)

• Occupation probabilities P1,2(t) for ω1 = ω⋆/2 and ω2 = 2 ∗ ω⋆

0 1 2 3 4 5 6 7t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P1

,2(t

)

Site occupation probabilities, 2

(t), =*/2

0 1 2 3 4 5 6 7t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P1

,2(t

)

Site occupation probabilities, 2

(t), =2**

• Plot of the phase function∫ t0 λ(t′) dt′

0 1 2 3 4 5 6 7t

0

2

4

6

8

10

12

14

16

18

0t

(t)

dt

Phase

=*

=*/2

=2**

Synchronizationπ

2=

∫ T/4

0λ(t) dt =

1

ω⋆

∫ π/2

0θ(s) ds ⇒ ω⋆ =

2

π

∫ π/2

0θ(s) ds ≈ 4.5


22Driven spin chain

Observations for N = 2:

➠ the transfer time form one end towards the other end of the chain is ωv-dependent,

however the excitation is always completely transferred (shorter times for ...);

➠ the time the excitation spans on the second site is also ωv-dependent, with a maximum for

a well-identified frequency ω⋆.

Driven model for N = 7:

dl(t) := d0 −[

z(m)l (t)− z

(m)l+1 (t)

]

, z(m)l (t) := d0 a sin

(

mπ l

N + 1

)

sin(ωv t+ ϕ) ,

λl(t) :=λ̃l

[dl(t)/d0]3, ∀t > 0 , l = 1, · · · , N − 1 ,

εl ≡ 0 , a ≡ 1/4 , d0 = λ̃l ≡ 1 , ϕ ≡ π/2 , ωv ∈ [2, 10] .

➠ mode m = 7: breathing mode; can be associated with a bucket brigade for firefighting,

transferring a water bucket from the reservoir towrads the fire.


23Driven sinusoidal model

• Excitation arrival time t⋆ at 7th site and P7(t⋆) for m = 7 (breathing mode) and several ωv

0 2 4 6 8 10v

1.5

2

2.5

3

3.5

4

4.5

t *

Excit. arrival time t* at N=7

0 2 4 6 8 10v

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P7(t

*)

Occupation probability P7

(t*)

• Excitation arrival time t⋆ at the 7th site and P7(t⋆) for m = 1 and several ωv

0 2 4 6 8 10v

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

5.1

t *

Excit. arrival time t* at N=7

0 2 4 6 8 10v

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

P7(t

*)

Occupation probability P7

(t*)


24Preliminar conclusions, second model

Observations:

➠ Pure quantum mechanical effect: the fact that a well-orchestrated time-dependent coupling

strength permits to enhance the excitation transfer as compared to a uniform coupling with

strength λavg;

➠ Concerted dynamics: in absence of synchronization between the vibrational motion and

the wave-like quantum excitation transfer, the pigments will not be able to transfer

efficiently the excitation towards the reaction center;

➠ Robustness: the coupling strength λ(t) enters into the computation of the site occupation

probability via and integration ⇔ small perturbations of these coupling coefficients, due

for example to environmental noise, will probably not be so dramatic, as compared to a

static coupling case.

Thank you for your attention !

enhanced excitation energy transferin the photosynthesis

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