Dynamical role of quantum coherence and environment in - PowerPoint PPT Presentation

Dynamical role of quantum coherence and environment in multichromophoric energy transfer Masoud Mohseni Patrick Rebentrost, Seth Lloyd, Alan Aspuru-Guzik Department of Chemistry and Chemical Biology, Harvard University Research Laboratory of Electronics , MIT

Quantum coherence in photosynthetic com plexes � G. Engel et al., Nature (07). � H. Lee, Y.-C. Cheng, G.R. Flemming, Science (07). P H O T O N S � E. Collini, G. Scholes, Science (09). � I, Mercer et al., Phys. Rev. Lett. (09). � How to explore dynamical interplay ANTENNA PIGMENTS between quantum coherence and (Chlorophyll molecules and other pigments) environment � How to quantify their contributions to energy transfer energy transfer efficiency P H O T O N S REACTION CENTER Fenna-Matthews-Olson (FMO) Complex

Partitioning Open Quantum Dynam ics Incoherent dynamics Coherent dynamics Quantum-classical regime Diffusion Unitary Unitary + relaxation + dephasing Statistical mixture of states Pure states Decohered states Can we partition dynamics of open quantum systems into contributions associated with fundamental physical mechanisms?

Multichrom ophoric Energy transfer A set of multichromophores which guides excitation energy between two points A and B. Bath Free Hamiltonian for M chromophores is: ( ) ∑ ∑ A B = ε + + † † † H a a V a a a a S m m m mn m n n m = < m 1 m n System-Bath Hamiltonians: Energy transfer channel = ∑ Thermal phonon bath † H q a a p m m m m Radiation field ∑ = + r † H q ( a a ) r m m m m

Lindblad m aster equation for m ultichrom ophoric system s Master Equation (Born-Markov and secular approximations): ∂ ρ ( ) t [ ] = − + ρ + ρ + ρ i H H , ( ) t L ( ) t L ( ) t ∂ S LS p r t Lindblad superoperator: ∑∑ 1 1 ρ = γ ω ω ρ ω − ω ω ρ − ρ ω ω k k k † k k † k k † L { ( )} t ( )[ A ( ) A ( ) A ( ) A ( ) A ( ) A ( )] k mn m n m n m n 2 2 ω m n , ∫ γ ω = ω i t ( ) C dte q ( ) t q (0) Bath correlations functions mn mn m n ( ) ( ) ( ) ω = * A c N c M M N Lindblad operators m MN m m 1 γ ω = π ω + ω + − ω − ω ω = ( ) 2 [ ( )(1 J n ( )) J ( ) ( n )] where n ( ) ω � − e KT 1 Reorganization energy ω E ( ) ω = ω − Bath spectral density: R J exp( ) ω ω � c c

Directed quantum walks in excitation manifolds Two-exciton manifold � 00 0 One-exciton manifold Directed quantum walk Quantum jumps in a fixed excitation manifold Damped evolution Ground state ∂ ρ ( ) t i ∗ ∑ ∑ ⎡ ⎤ = − ρ + Γ ρ + γ ρ p † r † H , ( ) t W W R R ⎣ ⎦ ∂ e f f m m n n , ', , ' m m , ' n n , ' m m m � t ∗ = [ ] − † † A B , AB B A Quantum jumps to j+1 or j-1 exciton manifold ∑ γ ρ r † Tr R [ R ] dt Probability of a jump: m m m = m 1 M Mohseni, P. Robentrost, S. Lloyd, A. Aspuru-Guzik , Journal of Chemical Physics (08) . A. Olaya-Castro, et. al, Phys. Rev. B (08)

Directed quantum w alks = + + H H H H eff S LS decoher N R i ∑ ∑∑ C = − Θ + γ ω + p † r † H [ a a ( ) a a H ] decoher m n , m n m m m trap 2 ω m m Classical transition matrix Classical Random Walk: dp t ( ) = ∑ a M p t ( ) ab b dt Transition (super-)matrix b Directed Quantum Walk: � � � � ρ d ( ) t ∑ = Μ ρ a ( ) t ab b dt b N i I ∑ C ∗ ∗ Μ = − ⊗ − ⊗ + Γ ⊗ p ( H H I ) ( W W ) ab eff eff m m n n , ', , ' n n , ' m m , ' ab � m m n n , ', , '

Energy Transfer Efficiency The energy transfer efficiency of the channel is Antenna defined as the integrated probability of the excitation successfully being trapped: = ∫ ∞ η ρ Tr H [ ( )] t dt trap 0 Loss The energy transfer time: = ∫ ∞ τ ρ tTr H [ ( )] t dt Reaction Center trap 0

Environm ent-assisted quantum w alks 1 3 6 T= 300K = 35cm -1 E R � 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)

3 W hy is environm ent helpful? 3 Barrier 1 6 1 Site basis 6 3 “Funnel” Energy basis 7 6 5 4 3 2 1

Partitioning Open quantum dynam ics Master equation (schematically) Coherent Relaxation Dephasing E i d dt ρ = ( ) t E j + + E E R R E k i H − � ρ Recom bination Trapping [ , ( )] t S + + RC P, Robentrost, M Mohseni, A. Aspuru-Guzik, H H J. Phys. Chem. B, in press (2009). recomb trap

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

Contributions to ETE = ∑ = + M L H L L ; k ref k Identity for Green’s function ∑ − − − − = + 1 1 1 1 M H H L M ref ref k k Coherent Relaxation Dephasing E i L = , E j , k E k Recom bination Trapping = H RC + ref 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).

Contributions to ETE = ∑ η η k k Simplify efficiency { } 2 Tr ( ) − η = − � ρ 1 H M 0 trap 2 η = − − Μ − ρ 1 1 Tr H { L (0)} k ref ' k �

Contributions to ETE relaxation coherent dephasing T= 300K = 35cm -1 E R � Crossover from quantum to relaxation regime � Quantum coherent contribution ~ 10% 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).

R → ∞ 0 dephasing relaxation R → coherent c c Spatially Correlated Bath mn δ 1 c R mn R DFS − e = mn C ; ( ) (0) q t q mn C = (0) n t q ( ) m q

dephasing relaxation coherent The effects of static disorder

Nonlocal dynam ical contributions to ETE ρ = ρ ( ) t F t ( ,0) (0) = ∑ M M k k → λ M M k k k ∂ ρ � � ∂ ρ ( ) t ( ) t 1 = ρ ∑ ∫ t = ρ M ( ) t F t t M F t ( , ') ( ',0) (0) dt ' ∂ ∂ t k t t 0 k ∞ 2 1 1 ∫ ∫ t η = ρ dt dt ' Tr H { ( )} t ∂ λ k trap � ± t ' 0 0 k ±

Energy transfer susceptibilities Com parison Green function’s method

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

Applications � 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?

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