Coherent Energy Transfer and Trapping on Networks (Quantum - - PowerPoint PPT Presentation

Coherent Energy Transfer and Trapping

• n Networks (Quantum Aggregates)

Oliver Mülken

QuEBS - Lisbon, June 7, 2009

Motivation: Light harvesting

[from Q. Rev. Biophys. 35, 1 (2002)] [from Wikipedia.org]

# Large variety of light harvesting complexes of well defined structure # Light is caputured by molecules

Motivation: Light harvesting

[from Q. Rev. Biophys. 35, 1 (2002)] [from Wikipedia.org]

# 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

Motivation: Light harvesting

[from Q. Rev. Biophys. 35, 1 (2002)] [from Wikipedia.org]

# 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 # Complex molecule mapped to network # Dynamics on networks => Study effect of coherence

Models for Transport

# Reduce complex system to “simple” network

from PRB 78, 085115 (2008) from New J. Phys. 11, 033003 (2009)

# Network: Collection of N participating centers (atoms, molecules, monomers, etc.)

Models for Transport

# Reduce complex system to “simple” network

from PRB 78, 085115 (2008) from New J. Phys. 11, 033003 (2009)

# Network: Collection of N participating centers (atoms, molecules, monomers, etc.)

* Interactions specify topology given by connectivity matrix A e.g.:

Akj = 8 > < > : fj for k = j −1 if k and j connected else,

* Includes possible disorder * Dynamics follows from topology

Models for Transport

# Reduce complex system to “simple” network

from PRB 78, 085115 (2008) from New J. Phys. 11, 033003 (2009)

# Network: Collection of N participating centers (atoms, molecules, monomers, etc.)

* Interactions specify topology given by connectivity matrix A e.g.:

Akj = 8 > < > : fj for k = j −1 if k and j connected else,

* Includes possible disorder * Dynamics follows from topology

# Incoherent/Coherent/Crossover aspects taking into account by

* Random Walks * Quantum Walks * Quantum Master Eqs.

Methods: CTRW - CTQW - QME

# Continuous-time random walks (CTRW)

• bey Master equation

d dtpk,j(t) =

• l

Tkl pl,j(t) Simplest case: transfer matrix T = A

Methods: CTRW - CTQW - QME

# Continuous-time random walks (CTRW)

• bey Master equation

d dtpk,j(t) =

• l

Tkl pl,j(t) Simplest case: transfer matrix T = A # Continuous-time quantum walks (CTQW)

• bey Schrödinger’s equation

d dtαk,j(t) = −i

• l

Hkl αl,j(t) Similar structure as CTRW

πk,j(t) ≡ |αk,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(Lj)

* “coupling operators” Lj 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 Γm|mm|

Γm ≡ Γ > 0 (m ∈ M ⊂ {1, . . . , N})

H = H0 − iΓ Γ Γ H = H0 − iΓ Γ Γ H = H0 − iΓ Γ Γ and T = T0 − Γ Γ Γ

H0/T0: 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 Γm|mm|

Γm ≡ Γ > 0 (m ∈ M ⊂ {1, . . . , N})

H = H0 − iΓ Γ Γ H = H0 − iΓ Γ Γ H = H0 − iΓ Γ Γ and T = T0 − Γ Γ Γ

H0/T0: without trapping

# Properties of H:

* Non-hermitian H = H† with El = ǫl − iγl El = ǫl − iγl El = ǫl − iγl and left (right) eigenstates

˜ Ψl|Ψl′ = δll′ and X

l

|Ψl ˜ Ψl| = 1 1 1

# γl determine the temporal decay of πk,j(t) = |αk,j(t)|2 αk,j(t) =

• l

exp[−γlt] exp[−γlt] exp[−γlt] exp[−iǫlt]k|Ψl˜ Ψl|j

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 ΠM(t) ≡ 1 N − M

• j∈M
• k∈M

πk,j(t) ΠM(t) ≡ 1 N − M

• j∈M
• k∈M

πk,j(t) ΠM(t) ≡ 1 N − M

• j∈M
• k∈M

πk,j(t) and PM(t) ≡ 1 N − M

• j∈M
• k∈M

pk,j(t)

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 ΠM(t) ≡ 1 N − M

• j∈M
• k∈M

πk,j(t) ΠM(t) ≡ 1 N − M

• j∈M
• k∈M

πk,j(t) ΠM(t) ≡ 1 N − M

• j∈M
• k∈M

πk,j(t) and PM(t) ≡ 1 N − M

• j∈M
• k∈M

pk,j(t) * For long t and small M/N, ΠM(t) simplifies: ΠM(t) ≈ 1 N − M

• l

exp[−2γlt] * Asymptotically, ΠM(t) dominated γmin: ΠM(t) = exp(−2γmint) ΠM(t) = exp(−2γmint) ΠM(t) = exp(−2γmint) * At intermediate times: γl often scale for some l: γl ∼ alµ ΠM(t) ∼ t−1/µ ΠM(t) ∼ t−1/µ ΠM(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

H0(ν) =

N

X

n=1

" n−1 X

R=1

R−ν“ |nn| − |n − Rn| ” +

N−n

X

R=1

R−ν“ |nn| − |n + Rn| ”#

γl ∼ l2 (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

H0(ν) =

N

X

n=1

" n−1 X

R=1

R−ν“ |nn| − |n − Rn| ” +

N−n

X

R=1

R−ν“ |nn| − |n + Rn| ”#

γl ∼ l2 (l ≪ N) for NNI * Slower decay due to LRI * 1st order pert.theory: γl ≈ γ(0)

l

+ 2−νγ(1)

l

µ ≈ µ(0) + 2−νµ(1) (µ(1) > 0)

ΠM (t) and PM (t) for N = 100 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: mj):

* Eigenstates without traps: Bloch states * Γ ≪ 1: 1st order corrections to eigenvalues E(1)

l

= γl

γN = 4 − iΓM/N, γN/2 = −iΓM/N, and γl = −iΓ/N “ M ± ˛ ˛ ˛

M

X

j=1

exp(i4πlmj/N) ˛ ˛ ˛ ”

Excitation trapping with LRI

[Int. J. Bif. Chaos, in press (2006)]

# Example: Ring (N even) with several traps Regular arrangement of traps (positions: mj):

* Eigenstates without traps: Bloch states * Γ ≪ 1: 1st order corrections to eigenvalues E(1)

l

= γl

γN = 4 − iΓM/N, γN/2 = −iΓM/N, and γl = −iΓ/N “ M ± ˛ ˛ ˛

M

X

j=1

exp(i4πlmj/N) ˛ ˛ ˛ ”

* Periodic distribution of traps, i.e., mj = jN/M with N/M ∈ N κ: nr. of l-values for which 2l/M ∈ N, then

˛ ˛ ˛

M

X

j=1

exp(i4πlj/M) ˛ ˛ ˛ = M

* zero contribution of mj 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: mj):

* Eigenstates without traps: Bloch states * Γ ≪ 1: 1st order corrections to eigenvalues E(1)

l

= γl

γN = 4 − iΓM/N, γN/2 = −iΓM/N, and γl = −iΓ/N “ M ± ˛ ˛ ˛

M

X

j=1

exp(i4πlmj/N) ˛ ˛ ˛ ”

* Periodic distribution of traps, i.e., mj = jN/M with N/M ∈ N κ: nr. of l-values for which 2l/M ∈ N, then

˛ ˛ ˛

M

X

j=1

exp(i4πlj/M) ˛ ˛ ˛ = M

* zero contribution of mj to (some) eigenstates

• > No decay of ΠM(t) to zero, but lim

t→∞ ΠM(t) =

κ 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 ±

• 1 − Γ2/4 − iΓ/2

E± = 1 ±

• 1 − Γ2/4 − iΓ/2

E± = 1 ±

• 1 − Γ2/4 − iΓ/2

* Survival probability Π(t) = π1,1(t) = e−Γt cos2 t

• 1 − Γ2/4
• Π(t) = π1,1(t) = e−Γt cos2

t

• 1 − Γ2/4
• Π(t) = π1,1(t) = e−Γt cos2

t

• 1 − Γ2/4

Coupling to an environment

[in preparation (2009)]

# Example: Dimer in a heat bath ⇆

* Two nodes (, ), one of which being the trap Eigenvalues: E± = 1 ±

• 1 − Γ2/4 − iΓ/2

E± = 1 ±

• 1 − Γ2/4 − iΓ/2

E± = 1 ±

• 1 − Γ2/4 − iΓ/2

* Survival probability Π(t) = π1,1(t) = e−Γt cos2 t

• 1 − Γ2/4
• Π(t) = π1,1(t) = e−Γt cos2

t

• 1 − Γ2/4
• Π(t) = π1,1(t) = e−Γt cos2

t

• 1 − Γ2/4
• * QME with trapping:

˙ ρ = −i

• H0, ρ
• −
• Γ, ρ
• − D(Lj)

Lj = λ|jj|

Coupling to an environment

[in preparation (2009)]

# Example: Dimer in a heat bath ⇆

* Two nodes (, ), one of which being the trap Eigenvalues: E± = 1 ±

• 1 − Γ2/4 − iΓ/2

E± = 1 ±

• 1 − Γ2/4 − iΓ/2

E± = 1 ±

• 1 − Γ2/4 − iΓ/2

* Survival probability Π(t) = π1,1(t) = e−Γt cos2 t

• 1 − Γ2/4
• Π(t) = π1,1(t) = e−Γt cos2

t

• 1 − Γ2/4
• Π(t) = π1,1(t) = e−Γt cos2

t

• 1 − Γ2/4
• * QME with trapping:

˙ ρ = −i

• H0, ρ
• −
• Γ, ρ
• − D(Lj)

Lj = λ|jj| * Solution for dimer for Γ = 0: π1,1(t) = 1 2 + e−λt 2

• λ sin
• t

√ 4 − λ2 √ 4 − λ2 cos

• t
• 4 − λ2

[recovers fully coherent solution for λ → 0]

Coupling to an environment

[in preparation (2009)]

# Dimer in a heat bath

* Approximate solution for Γ = 0 by expanding to first order in Γ and λ [Eq.(14)]: Π(t) ≈ e−Γt1 2 + e−λt 2

• cos 2t + λ

2 sin 2t

• Π(t) ≈ e−Γt1

2 + e−λt 2

• cos 2t + λ

2 sin 2t

• Π(t) ≈ e−Γt1

2 + e−λt 2

• cos 2t + λ

2 sin 2t

Coupling to an environment

[in preparation (2009)]

# Dimer in a heat bath

* Approximate solution for Γ = 0 by expanding to first order in Γ and λ [Eq.(14)]: Π(t) ≈ e−Γt1 2 + e−λt 2

• cos 2t + λ

2 sin 2t

• Π(t) ≈ e−Γt1

2 + e−λt 2

• cos 2t + λ

2 sin 2t

• Π(t) ≈ e−Γt1

2 + e−λt 2

• cos 2t + λ

2 sin 2t

• Π(t) for λ = 0.01 and Γ = 0.01

Comparison to Path-Integral Monte Carlo calculations

Coupling to an environment

[in preparation (2009)]

# Example: Trimer in a heat bath

• ւ

տ

←

* Three nodes (, , ), one of which being the trap . * Without heat bath: lim

t→∞ Π(t) = N − 2

N − 1 = 1 2

Coupling to an environment

[in preparation (2009)]

# Example: Trimer in a heat bath

• ւ

տ

←

* Three nodes (, , ), one of which being the trap . * Without heat bath: lim

t→∞ Π(t) = N − 2

N − 1 = 1 2 * Heat bath destroys effect of nonvanishing survival probability

Π(t) for λ = 0.01 and Γ = 0.1 Π(t) for λ = 0.1 and Γ = 0.01

Conclusions

# CTQW, CTRW, and crossover dynamics on networks # Excitation trapping

• > Unexpected behavior for trapping with LRI
• > Finite survival probability Π(t) for excitations on rings

# Coupling to external heat bath

• > Dimer: Exact solution agrees with PIMC
• > Trimer: Coupling destroy effect of finite Π(t)

# Experimental realization based on Rydberg gases in preparation (Matthias Weidemüller’s group in Heidelberg)

Conclusions

# CTQW, CTRW, and crossover dynamics on networks # Excitation trapping

• > Unexpected behavior for trapping with LRI
• > Finite survival probability Π(t) for excitations on rings

# Coupling to external heat bath

• > Dimer: Exact solution agrees with PIMC
• > Trimer: Coupling destroy effect of finite Π(t)

# Experimental realization based on Rydberg gases in preparation (Matthias Weidemüller’s group in Heidelberg)

Take-home messages

# Complex molecules can be mapped on “simple” networks # Mathematical models include only network’s topology # Some essential feature of transport processes caputered by simple models

Final statements . . .

Thanks to Volker Pernice Elena Agliari Alexander Blumen

(CTQW on networks)

Lothar Mühlbacher

(PIMC)

Thomas Amthor Christian Giese Markus Reetz-Lamour Christoph Hofmann Matthias Weidemüller

(Rydberg gases experiments)

“If that turns out to be true, I’ll quit physics.” (Max von Laue)

talking about electrons having wave properties

“Very interesting theory - it makes no sense at all.” (Groucho Marx) “I think it is safe to say that no one understands quantum mechanics.” (Richard P. Feynman)