DAY 1. Photosynthesis Photosynthesis Song Photosynthesis Song.
Enhanced Excitation Energy Transferin the photosynthesis ...
Transcript of Enhanced Excitation Energy Transferin the photosynthesis ...
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C. Negulescu, march 2021
1
Enhanced Excitation Energy Transfer in the
photosynthesis process
Claudia Negulescu
Institut de Mathématiques de Toulouse
Université Paul Sabatier
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C. Negulescu, march 2021
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?
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C. Negulescu, march 2021
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
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C. Negulescu, march 2021
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!
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C. Negulescu, march 2021
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?
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C. Negulescu, march 2021
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!
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C. Negulescu, march 2021
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.
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C. Negulescu, march 2021
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
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C. Negulescu, march 2021
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.
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C. Negulescu, march 2021
10
The photosynthesis process
First mathematical model
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C. Negulescu, march 2021
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
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C. Negulescu, march 2021
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
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C. Negulescu, march 2021
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
Occupation probability ||j(t)||
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
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C. Negulescu, march 2021
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
Occupation probability ||j(t)||
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
Occupation probability ||j(t)||
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?
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C. Negulescu, march 2021
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
Occupation probability ||j(t)||
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
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C. Negulescu, march 2021
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)
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C. Negulescu, march 2021
17
The photosynthesis process
Second mathematical model
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C. Negulescu, march 2021
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
]
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C. Negulescu, march 2021
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′)
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C. Negulescu, march 2021
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)
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C. Negulescu, march 2021
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
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C. Negulescu, march 2021
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.
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C. Negulescu, march 2021
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*)
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C. Negulescu, march 2021
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 !