Coherent Energy Transfer and Trapping on Networks (Quantum Aggregates) Oliver Mülken QuEBS - Lisbon, June 7, 2009
Motivation: Light harvesting # Large variety of light harvesting complexes of well defined structure # Light is caputured by molecules [from Q. Rev. Biophys. 35, 1 (2002)] [from Wikipedia.org]
Motivation: Light harvesting # Large variety of light harvesting complexes of well defined structure # Light is caputured by molecules # Excitation energy transport # Transfer to reaction center (RC) # Very fast and highly efficient process Not describable by diffusive process [from Q. Rev. Biophys. 35, 1 (2002)] [from Wikipedia.org]
Motivation: Light harvesting # Large variety of light harvesting complexes of well defined structure # Light is caputured by molecules # Excitation energy transport # Transfer to reaction center (RC) # Very fast and highly efficient process Not describable by diffusive process [from Q. Rev. Biophys. 35, 1 (2002)] # Complex molecule mapped to network # Dynamics on networks => Study effect of coherence [from Wikipedia.org]
Models for Transport # Reduce complex system to “simple” network # Network: Collection of N participating centers (atoms, molecules, monomers, etc.) from PRB 78, 085115 (2008) from New J. Phys. 11, 033003 (2009)
Models for Transport # Reduce complex system to “simple” network # Network: Collection of N participating centers (atoms, molecules, monomers, etc.) * Interactions specify topology given by connectivity matrix A 8 for k = j f j e.g.: > < A kj = − 1 if k and j connected from PRB 78, 085115 (2008) > else , 0 : * Includes possible disorder * Dynamics follows from topology from New J. Phys. 11, 033003 (2009)
Models for Transport # Reduce complex system to “simple” network # Network: Collection of N participating centers (atoms, molecules, monomers, etc.) * Interactions specify topology given by connectivity matrix A 8 for k = j f j e.g.: > < A kj = − 1 if k and j connected from PRB 78, 085115 (2008) > else , 0 : * Includes possible disorder * Dynamics follows from topology # Incoherent/Coherent/Crossover aspects taking into account by * Random Walks * Quantum Walks * Quantum Master Eqs. from New J. Phys. 11, 033003 (2009)
Methods: CTRW - CTQW - QME # Continuous-time random walks (CTRW) d � obey Master equation dtp k,j ( t ) = T kl p l,j ( t ) l Simplest case: transfer matrix T = A
Methods: CTRW - CTQW - QME # Continuous-time random walks (CTRW) d � obey Master equation dtp k,j ( t ) = T kl p l,j ( t ) l Simplest case: transfer matrix T = A # Continuous-time quantum walks (CTQW) d � obey Schrödinger’s equation dtα k,j ( t ) = − i H kl α l,j ( t ) l Similar structure as CTRW π k,j ( t ) ≡ | α k,j ( t ) | 2 -> identify Hamiltonian H with T [Farhi, Gutmann - PRA 58 (1998)] => Comparison of incoherent and coherent dynamics
Methods: CTRW - CTQW - QME # CTRW-CTQW Crossover * start from QW -> reformulate for density operator ρ -> Liouville-von-Neumann equation (LvNE) � � ρ = − i ˙ H , ρ
Methods: CTRW - CTQW - QME # CTRW-CTQW Crossover * start from QW -> reformulate for density operator ρ -> Liouville-von-Neumann equation (LvNE) � � ρ = − i ˙ H , ρ * coupling to “environment” (heat bath) * Lindblad formalism for coupling -> new equation (QME) � � ρ = − i ˙ H , ρ − D ( L j ) * “coupling operators” L j model environment => decoherence/dephasing effects
Excitation trapping [PRL 99, 090601 (2007); PRE 78, 021115 (2008); arXiv:0810.4052 (2008)] # Out of the N nodes M are traps with M ≤ N (“absorbing states”, ”reaction centers”, etc.) # Phenomenological trapping operator Γ Γ ≡ � Γ m Γ m | m �� m | Γ m ≡ Γ > 0 ( m ∈ M ⊂ { 1 , . . . , N } ) Γ and H = H 0 − i Γ H = H 0 − i Γ H = H 0 − i Γ Γ Γ Γ Γ Γ T = T 0 − Γ Γ Γ H 0 / T 0 : without trapping
Excitation trapping [PRL 99, 090601 (2007); PRE 78, 021115 (2008); arXiv:0810.4052 (2008)] # Out of the N nodes M are traps with M ≤ N (“absorbing states”, ”reaction centers”, etc.) # Phenomenological trapping operator Γ Γ ≡ � Γ m Γ m | m �� m | Γ m ≡ Γ > 0 ( m ∈ M ⊂ { 1 , . . . , N } ) Γ and H = H 0 − i Γ H = H 0 − i Γ H = H 0 − i Γ Γ Γ Γ Γ Γ T = T 0 − Γ Γ Γ H 0 / T 0 : without trapping # Properties of H : * Non-hermitian H � = H † with E l = ǫ l − iγ l E l = ǫ l − iγ l E l = ǫ l − iγ l and left (right) eigenstates � ˜ X | Ψ l �� ˜ Ψ l | Ψ l ′ � = δ ll ′ and Ψ l | = 1 1 1 l # γ l determine the temporal decay of π k,j ( t ) = | α k,j ( t ) | 2 exp[ − γ l t ] exp[ − iǫ l t ] � k | Ψ l �� ˜ � α k,j ( t ) = exp[ − γ l t ] exp[ − γ l t ] Ψ l | j � l
Excitation trapping [PRL 99, 090601 (2007); PRE 78, 021115 (2008); arXiv:0810.4052 (2008)] # Keep track of all nodes k �∈ M * Calculate mean survival probabilities 1 1 1 1 � � � � � � � � π k,j ( t ) and P M ( t ) ≡ Π M ( t ) ≡ Π M ( t ) ≡ Π M ( t ) ≡ π k,j ( t ) π k,j ( t ) p k,j ( t ) N − M N − M N − M N − M j �∈M j �∈M j �∈M k �∈M k �∈M k �∈M j �∈M k �∈M
Excitation trapping [PRL 99, 090601 (2007); PRE 78, 021115 (2008); arXiv:0810.4052 (2008)] # Keep track of all nodes k �∈ M * Calculate mean survival probabilities 1 1 1 1 � � � � � � � � π k,j ( t ) and P M ( t ) ≡ Π M ( t ) ≡ Π M ( t ) ≡ Π M ( t ) ≡ π k,j ( t ) π k,j ( t ) p k,j ( t ) N − M N − M N − M N − M j �∈M j �∈M j �∈M k �∈M k �∈M k �∈M j �∈M k �∈M * For long t and small M/N , Π M ( t ) simplifies: 1 � Π M ( t ) ≈ exp[ − 2 γ l t ] N − M l * Asymptotically, Π M ( t ) dominated γ min : Π M ( t ) = exp( − 2 γ min t ) Π M ( t ) = exp( − 2 γ min t ) Π M ( t ) = exp( − 2 γ min t ) * At intermediate times: γ l often scale for some l : γ l ∼ al µ Π M ( t ) ∼ t − 1 /µ Π M ( t ) ∼ t − 1 /µ Π M ( t ) ∼ t − 1 /µ
Excitation trapping with LRI [PRE 78, 021115 (2008)] # Example: Chain with traps at both ends ( Γ ≪ 1 ) * LRI with R − ν = | k − j | − ν " n − 1 N R − ν “ ” X X H 0 ( ν ) = | n �� n | − | n − R �� n | n =1 R =1 N − n ”# R − ν “ X + | n �� n | − | n + R �� n | R =1 γ l ∼ l 2 ( l ≪ N ) for NNI
Excitation trapping with LRI [PRE 78, 021115 (2008)] # Example: Chain with traps at both ends ( Γ ≪ 1 ) * LRI with R − ν = | k − j | − ν " n − 1 N R − ν “ ” X X H 0 ( ν ) = | n �� n | − | n − R �� n | n =1 R =1 N − n ”# R − ν “ X + | n �� n | − | n + R �� n | R =1 γ l ∼ l 2 ( l ≪ N ) for NNI * Slower decay due to LRI * 1 st order pert.theory: γ l ≈ γ (0) + 2 − ν γ (1) Π M ( t ) and P M ( t ) for N = 100 l l µ ≈ µ (0) + 2 − ν µ (1) ( µ (1) > 0 ) with (a) Γ = 0 . 001 and (b) Γ = 1
Excitation trapping with LRI [Int. J. Bif. Chaos, in press (2006)] # Example: Ring ( N even) with several traps Regular arrangement of traps (positions: m j ): * Eigenstates without traps: Bloch states * Γ ≪ 1 : 1st order corrections to eigenvalues E (1) = γ l l M “ ˛ ˛ ” X γ N = 4 − i Γ M/N , γ N/ 2 = − i Γ M/N , and γ l = − i Γ /N M ± exp( i 4 πlm j /N ) ˛ ˛ ˛ ˛ j =1
Excitation trapping with LRI [Int. J. Bif. Chaos, in press (2006)] # Example: Ring ( N even) with several traps Regular arrangement of traps (positions: m j ): * Eigenstates without traps: Bloch states * Γ ≪ 1 : 1st order corrections to eigenvalues E (1) = γ l l M “ ˛ ˛ ” X γ N = 4 − i Γ M/N , γ N/ 2 = − i Γ M/N , and γ l = − i Γ /N M ± exp( i 4 πlm j /N ) ˛ ˛ ˛ ˛ j =1 * Periodic distribution of traps, i.e., m j = jN/M with N/M ∈ N M κ : nr. of l -values for which 2 l/M ∈ N , then ˛ ˛ X exp( i 4 πlj/M ) ˛ = M ˛ ˛ ˛ j =1 * zero contribution of m j to (some) eigenstates
Excitation trapping with LRI [Int. J. Bif. Chaos, in press (2006)] # Example: Ring ( N even) with several traps Regular arrangement of traps (positions: m j ): * Eigenstates without traps: Bloch states * Γ ≪ 1 : 1st order corrections to eigenvalues E (1) = γ l l M “ ˛ ˛ ” X γ N = 4 − i Γ M/N , γ N/ 2 = − i Γ M/N , and γ l = − i Γ /N M ± exp( i 4 πlm j /N ) ˛ ˛ ˛ ˛ j =1 * Periodic distribution of traps, i.e., m j = jN/M with N/M ∈ N M κ : nr. of l -values for which 2 l/M ∈ N , then ˛ ˛ X exp( i 4 πlj/M ) ˛ = M ˛ ˛ ˛ j =1 * zero contribution of m j to (some) eigenstates κ -> No decay of Π M ( t ) to zero, but lim t →∞ Π M ( t ) = N − M
Coupling to an environment [in preparation (2009)] # Example: Dimer in a heat bath � ⇆ � * Two nodes ( � , � ), one of which being the trap � � Eigenvalues: E ± = 1 ± � � E ± = 1 ± E ± = 1 ± 1 − Γ 2 / 4 − i Γ / 2 1 − Γ 2 / 4 − i Γ / 2 1 − Γ 2 / 4 − i Γ / 2 Π( t ) = π 1 , 1 ( t ) = e − Γ t cos 2 � Π( t ) = π 1 , 1 ( t ) = e − Γ t cos 2 � Π( t ) = π 1 , 1 ( t ) = e − Γ t cos 2 � � � � � � � * Survival probability 1 − Γ 2 / 4 t t t 1 − Γ 2 / 4 1 − Γ 2 / 4
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