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Dynamical role of quantum coherence and environment in multichromophoric energy transfer

Department of Chemistry and Chemical Biology, Harvard University Research Laboratory of Electronics, MIT

Masoud Mohseni

Patrick Rebentrost, Seth Lloyd, Alan Aspuru-Guzik

P H O T O N S

ANTENNA PIGMENTS

(Chlorophyll molecules and other pigments) energy transfer

P H O T O N S

REACTION CENTER

Quantum coherence in photosynthetic com plexes

Fenna-Matthews-Olson (FMO) Complex

• G. Engel et al., Nature (07).
• H. Lee, Y.-C. Cheng, G.R. Flemming, Science (07).
• E. Collini, G. Scholes, Science (09).

I, Mercer et al., Phys. Rev. Lett. (09).

How to explore dynamical interplay between quantum coherence and environment How to quantify their contributions to energy transfer efficiency

Can we partition dynamics of open quantum systems into contributions associated with fundamental physical mechanisms?

Coherent dynamics

Partitioning Open Quantum Dynam ics

Unitary Pure states Quantum-classical regime

Unitary + relaxation + dephasing Decohered states

Incoherent dynamics

Diffusion Statistical mixture of states

Multichrom ophoric Energy transfer

A set of multichromophores which guides excitation energy between two points A and B.

( )

† † † 1 S m m m mn m n n m m m n

H a a V a a a a ε

= <

= + +

∑ ∑

Free Hamiltonian for M chromophores is: Bath Energy transfer channel

A B

† p m m m m

H q a a =∑

†

( )

r r m m m m

H q a a = +

∑

System-Bath Hamiltonians: Thermal phonon bath Radiation field

[ ]

( ) , ( ) ( ) ( )

S LS p r

t i H H t L t L t t ρ ρ ρ ρ ∂ = − + + + ∂

Lindblad superoperator:

† † † ,

1 1 { ( )} ( )[ ( ) ( ) ( ) ( ) ( ) ( )] 2 2

k k k k k k k k mn m n m n m n m n

L t A A A A A A

ω

ρ γ ω ω ρ ω ω ω ρ ρ ω ω = − −

∑∑ Lindblad m aster equation for m ultichrom ophoric system s

Master Equation (Born-Markov and secular approximations): ( ) 2 [ ( )(1 ( )) ( ) ( )] J n J n γ ω π ω ω ω ω = + + − −

( ) ( ) (0)

i t mn mn m n

C dte q t q

ω

γ ω =

∫

Bath correlations functions Lindblad operators

( ) ( ) ( )

* m MN m m

A c N c M M N ω =

Bath spectral density:

( )

exp( )

R c c

E J ω ω ω ω ω = −

• Reorganization energy

1 ( ) 1

KT

where n e

ω

ω = −

Directed quantum walk

Ground state One-exciton manifold Two-exciton manifold

Directed quantum walks in excitation manifolds

Quantum jumps in a fixed excitation manifold Damped evolution

00

• M Mohseni, P. Robentrost, S. Lloyd, A. Aspuru-Guzik, Journal of Chemical Physics (08).
• A. Olaya-Castro, et. al, Phys. Rev. B (08)

[ ]

† †

, A B AB B A

∗ =

−

Quantum jumps to j+1 or j-1 exciton manifold Probability of a jump:

† 1

[ ]

r m m m m

Tr R R dt γ ρ

=

∑

† † , ', , ' , ' , '

( ) , ( )

p r m m n n m m f n n f m m e m

t i W H t W t R R ρ γ ρ ρ ρ

∗

∂ ⎡ ⎤ = − + Γ + ⎣ ⎦ ∂

∑ ∑

† † ,

[ ( ) ] 2

C

N R p r decoher m n m n m m m trap m m

i H a a a a H

ω

γ ω = − Θ + +

∑ ∑∑

Directed quantum w alks

eff S LS decoher

H H H H = + +

( ) ( )

a ab b b

dp t M p t dt =∑ ( ) ( )

a b ab b

d t t dt ρ ρ = Μ

∑

• Classical Random Walk:

Directed Quantum Walk: Transition (super-)matrix Classical transition matrix

, ', , ' , ' , ' , ', , '

( ) ( )

C

N p ab eff eff m m n n n n m m ab m m n n

i I H H I W W

∗ ∗

Μ = − ⊗ − ⊗ + Γ ⊗

∑

Energy Transfer Efficiency

[ ( )]

trap

Tr H t dt η ρ

∞

= ∫

The energy transfer efficiency of the channel is defined as the integrated probability of the excitation successfully being trapped: Antenna Loss Reaction Center

[ ( )]

trap

tTr H t dt τ ρ

∞

= ∫

The energy transfer time:

Environm ent-assisted quantum w alks

Why improvement is helpful? [ Easy] How to quantify the role of coherence/ environment. [ Not so easy]

Masoud Mohseni, P. Robentrost, Seth Lloyd, Alan Aspuru-Guzik,, Journal of Chemical Physics 129, 174106 (2008)

T= 300K ER = 35cm -1

1 6 3

W hy is environm ent helpful?

1 6 3 3 1

Barrier

6 3

7 6 5 4 3 2 1 “Funnel” Energy basis Site basis

Partitioning Open quantum dynam ics

Ei Ej

+ +

Coherent Relaxation Dephasing Ek Recom bination Trapping RC

+ +

[ , ( )]

S

i H t ρ −

trap

H

recomb

H ( ) d t dt ρ =

R

E

R

E

Master equation (schematically)

P, Robentrost, M Mohseni, A. Aspuru-Guzik,

• J. Phys. Chem. B, in press (2009).

+ +

Contributions to efficiency Master equation Efficiency Superoperator For example What is the contribution of coherent evolution?

Partitioning Energy Transfer Efficiency

1 1 1 1 ref ref k k

M H H L M

− − − −

= +

∑

Ei Ej Relaxation Ek Dephasing Coherent

k

L =

ref

H =

Recom bination Trapping RC Identity for Green’s function

Sim ilar in spirit to M.K. Sener,… ,K. Schulten, J Chem. Phys. 120, 11183 (2004). J.A. Leegwater, J. Phys. Chem. 100, 14403 (1996).

ref

M L H = +

k k

L L = ∑

; + , ,

Contributions to ETE

k k

η η =∑

1 1 '

2 { (0)}

k ref k

Tr H L η ρ

− −

= − Μ

• ( )

{ }

1

2 Tr

trap

H M η ρ

−

= −

Simplify efficiency

Contributions to ETE

coherent relaxation dephasing Crossover from quantum to relaxation regime Quantum coherent contribution ~ 10% T= 300K ER = 35cm -1

P, Robentrost, M Mohseni, A. Aspuru-Guzik, Role of Quantum Coherence and Environmental Fluctuations in Chromophoric Energy Transport, J. Phys. Chem. B, in press (2009).

Contributions to ETE

Spatially Correlated Bath

( ) (0) ( ) (0)

m n mn

q t q C q t q =

mn c

R R mn

C e

−

=

;

coherent relaxation dephasing

mn

δ

1

c

R →

c

R → ∞

DFS

The effects of static disorder

coherent relaxation dephasing

2 1 1 ' { ( )} '

t k trap k

dt dt Tr H t t η ρ λ

± ±

∞

= ∂

∫ ∫

• ( )

( ,0) (0) t F t ρ ρ =

k k

M M =∑

Nonlocal dynam ical contributions to ETE

( ) 1 ( , ') ( ',0) (0) '

t k k

t F t t M F t dt t t ρ ρ ∂ = ∂

∑ ∫

• ( )

( ) t M t t ρ ρ ∂ = ∂

k k k

M M λ →

Green function’s method Energy transfer susceptibilities

Com parison

Spatial Pathw ays

Conclusion & Outlook

• Environm ental interactions lead to high energy

transfer efficiency for the FMO com plex

• Contribution m easure reveals underlying dynam ics
• FMO com plex: Relaxation is dom inant effects ~80%,

quantum coherence ~ 1 0 %

Generalization to non-Markovian dynam ics and

strong bath Quantifying the lim itations of quantum transport in open system s Optim izing charge and energy transfer

How can we improve quantum state transfer in noisy and disordered networks? Can we enhance charge and energy transfer in biological systems and nano-devices using quantum interference? Can we engineer artificial materials to achieve optimal energy transport in realistic environments by exploiting quantum effects?

Applications

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