Coherent - incoherent transitions in resonant energy transfer - - PowerPoint PPT Presentation

Coherent - incoherent transitions in resonant energy transfer dynamics

Ahsan Nazir

EPSRC Postdoctoral Fellow, University College London Centre for Quantum Dynamics, Griffith University, Australia AN, arXiv:0906.0592

Outline

• Explore criteria for coherent or incoherent energy transfer in a donor-acceptor

pair, beyond weak system-bath coupling

• Analytical theory: polaron transformation + time-local master equation
• Crossover from coherent to incoherent dynamics with increasing temperature:

multi-phonon dephasing effects begin to dominate

• Crossover temp. Tc displays pronounced dependence on the degree of

correlation between fluctuations at donor and acceptor sites

• Strong correlations suppression of multi-phonon processes: coherent

dynamics can then survive at elevated temperatures

Motivation - Quantum dots

• See also Crooker et al.

PRL 89, 186802 (2002), and Kagan et al. PRB 54, 8633 (1996) Weak exciton-phonon coupling Incoherent transfer

Motivation

• Energy transfer is ubiquitous
• Experimental observation of

coherent effects

• Conditions for coherent or

incoherent transfer?

• How can coherence survive at

elevated temperatures?

• Can we understand observed

dynamics from simple models?

Conjugated Polymers: E. Collini and G. D. Scholes, Science 323, 369 (2009) LH1-RC: Fig. courtesy of A. Olaya-Castro QDs: B. D. Gerardot et al., Phys. Rev. Lett. 95, 137403 (2005) FMO: Fig. courtesy of Y.-C. Cheng and G. R. Fleming, Annu. Rev. Phys. Chem. 60, 241 (2009) , G. S. Engel et al., Nature 446, 782 (2007)

Model

• Non-perturbative in donor-

acceptor electronic couplings

• Weak - strong system bath

coupling (single to multi- phonon effects)

• Low - high temperatures
• Correlated - uncorrelated

dephasing fluctuations

• Polaron transformation

d

V

|X |X |0 |0

g1 g2

Environment

ǫ1

ǫ2

H = ǫ1|X1X| + ǫ2|X2X| + V (|0XX0| + |X00X|) +

• k

ωkb†

kbk

+|X1X|

• k

(gk,1b†

k + g∗ k,1bk) + |X2X|

• k

(gk,2b†

k + g∗ k,bk)

Donor Acceptor

gk,1 = |gk|eik·d/2 gk,2 = |gk|e−ik·d/2

Previous work

Foerster - Dexter: Strong system-bath coupling, weak donor-acceptor interactions Weak system-bath coupling: Coherence in Photosynthetic networks investigated using Lindblad master equations Extended to consider coherence effects within donors and acceptors

Previous works (II)

Non-Markovian dynamics Modified Redfield treatment Polaron transformation: Interpolates weak to strong system bath interactions

• Single excitation subspace, map to a 2-level system
• Polaron transformation
• Bath-renormalised interaction

Model Hamiltonian

Environment Effective 2LS

g1 − g2 |0X |X0

V

J(ω) = αω3e−(ω/ωc)2

F1D(ω, d) = cos (ωd/c)

F2D(ω, d) = J0(ωd/c)

F3D(ω, d) = sinc(ωd/c)

HP = esHe−s = ǫ 2σz + VRσx +

• k

ωkb†

kbk + V (σxBx + σyBy)

e±s = |X0X0|ΠkD(gk,1/ωk) + |0X0X|ΠkD(gk,2/ωk)

5 10 15 20 2 4 6 8 Ω ps JΩ1FΩ,d ps1

Effective spectral density

VR = BV = e−

R ∞ dω J(ω)

ω2 (1−F (ω,d)) coth ω/2kBT V

Hsub = ǫ1|00| + ǫ2|11| + V (|10| + |01|) +

• k

ωkb†

kbk

+(|00| −| 11|)

• k
• (gk,1 − gk,2)b†

k + (gk,1 − gk,2)∗bk

• |0

|1

Master equation

• Treat fluctuations to second order: Born-Markov approximation
• Rates are FTs of bath correlation functions
• In terms of phonon propagator

˙ ρ = −iη 2[σz, ρ] − V 2 2

• l,ω,ω′

((γl(ω′) + 2iSl(ω′))[Pl(ω), Pl(ω′)ρ] + H.c.) η =

• ǫ2 + 4V 2

R

ω, ω′ ∈ {0, ±η} γx/y(ω) = eω/kBT ∞

−∞

dτeiωτΛx/y(τ) Λy(τ) = B2 2 (e ¯

ϕ(τ) − e− ¯ ϕ(τ))

Λx(τ) = B2 2 (e ¯

ϕ(τ) + e− ¯ ϕ(τ) − 2),

5 10 x 20 10 10 20 Τ 0.0 0.5

• ¯

ϕ(τ) = 2 ∞ dω J(ω) ω2 (1 − F(ω, d)) cos ωτ sinh (ω/2kBT)

Resonant dynamics

• Initialise in donor, look at

subsequent excitation dynamics

• Coherent or incoherent

dynamics possible

σzt = e−(Γ1+Γ2)t/2

• cos ξt

2 + (Γ2 − Γ1) ξ sin ξt 2

• ξ ≈
• 16V 2

R − (Γ1 − Γ2)2

50 100 150 200 1.0 0.5 0.0 0.5 1.0 t Σzt 50 100 150 200 1.0 0.5 0.0 0.5 1.0 t Σzt 50 100 150 200 1.0 0.5 0.0 0.5 1.0 t Σzt

Γ1 = V 2

• 2γx(0) + γy(2VR)(1 + 2N(2VR))

(1 + N(2VR))

• ,

Γ2 = 2V 2γx(0), 4VR > (Γ1 − Γ2) → ξ real

Coherent

4VR = (Γ1 − Γ2) → ξ = 0

Crossover

4VR < (Γ1 − Γ2) → ξ imaginary

Incoherent

ǫ = 0

N(ω) = (eω/kBT − 1)−1

Weak coupling - coherent

• Expand bath correlation functions to first
• rder in to get single-phonon rates
• Damped oscillations
• is real
• Damping consistent with a weak-

coupling treatment

50 100 150 200 1.0 0.5 0.0 0.5 1.0 t Σzt

¯ ϕ(τ) Λx(τ) ≈ 0, Λy(τ) ≈ ˜ B2 ¯ ϕ(τ) σzt = e−˜

Γ1t/2[cos (˜

ξt/2) − (˜ Γ1/˜ ξ) sin (˜ ξt/2)] ˜ ξ =

• 16 ˜

V 2

R − ˜

Γ2

1

˜ Γ1 = πJ(2 ˜ VR)(1 − F(2 ˜ VR, d)) coth ( ˜ VR/kBT)

5 10 15 20 25 0.00 0.01 0.02 0.03 0.04 d nm

• 1 ps1

1D 2D 3D

When are single phonon rates valid?

• Example: consider a particular

spectral density

• This leads to:
• Two important temperature scales:
• Weak fluctuation correlations
• Strong fluctuation correlations

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0000 0.0005 0.0010 0.0015 0.0020 TT0 122 arb. units 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 TT0 122 arb. units

5 10 x 20 10 10 20 Τ 0.0 0.5

• J(ω) = αω3

¯ ϕ(τ ′) = ϕ0

• sech2τ ′ − tanh (x − τ ′) + tanh (x + τ ′)

2x

• ϕ0 = 2π2αk2

BT 2 = T 2/T 2 0 ,

x = πkBTd/c = T/Td x ≫ 1(T ≫ Td) → ϕ0 ≪ 1 x ≪ 1(T ≪ Td) → ϕ0x2 ≪ 1

Td small Td large

Incoherent - High temperature, multi-phonon

• Estimate rates by saddle-point

approximation (essentially expand about )

• For high enough temperature

we have incoherent transfer

• Considering multi-phonon

processes gives rise to a destruction of coherent effects

50 100 150 200 1.0 0.5 0.0 0.5 1.0 t Σzt

1 2 3 4 5 6 7 104 0.001 0.01 0.1 1 10 TT0 122 arb. units

¯ ϕ(τ) τ = 0

weak full saddle point

Γ1 ≈ 2Γ2 ≈ 2β V 2

F B2 0e2ϕ0/3eϕ0(2xcsch2x−1)/x2

• πϕ0(x − sech2x tanh x)/x

ξ ≈

• 16V 2

R − Γ2 1/4 ≈ iΓ1/2 → σzt ≈ e−Γ1t

Crossover

• Crossover generally occurs in

the high temperature regime.

• Simplified condition:
• Critical temperature Tc
• Strong correlations suppress

multi-phonon effects

1 2 3 4 2 4 6 8 10 dd0T0Td TcT0

Γ1 − Γ2 4VR

2 4 6 8 1 2 3 4 TT0

8VR = Γ1 8VR > Γ1 → coherent 8VR < Γ1 → incoherent ϕc = T 2

c /T 2 0 ,

T 2

c = T0

VF B0e5ϕc/6eϕc(coth xc−2 tanh xc−1/xc)/2xc 4kB

• π(xc − sech2xc tanh xc)/xc

,

xc = Tc/Td

Strong correlation weak correlation

Far off-resonance

• Pairs of QDs not usually naturally

resonant

• Expand to 2nd order in
• Incoherent transfer from donor to

acceptor, as expected

• Weak coupling (single-phonon):

comparison with Rozbicki and Machnikoski

• High-temperature:

V/ǫ ≪ 1

QDs: B. D. Gerardot et al., Phys. Rev. Lett. 95, 137403 (2005)

V/ǫ σzt ≈ e−Γt − (1 − e−Γt) tanh (βǫ/2)

Γ = V 2 (1 + 2N(ǫ)) (1 + N(ǫ)) (γx(ǫ) + γy(ǫ))

˜ Γ ≈ (4π ˜ V 2

R/ǫ2)J(ǫ)(1 − F(ǫ, d)) coth (βǫ/2)

σzt ≈ e−Γt

0.0 0.5 1.0 1.5 2.0 2.5 106 105 104 0.001 0.01 TT0 Inc arb. units

Summary and further work (AN, arXiv0906.0592)

• Explored criteria for coherent or incoherent energy transfer in a donor-

acceptor pair, beyond weak system-bath coupling

• Crossover from coherent to incoherent dynamics with increasing temperature:

multi-phonon dephasing effects begin to dominate

• Crossover temp. Tc displays pronounced dependence on the degree of

correlation between fluctuations at donor and acceptor sites

• Applications to real systems, in particular biological systems?
• Non-Markovian and initial state preparation effects (see S. Jang et al., J.
• Chem. Phys. 129, 101104 (2008))

• Funding

Acknowledgements

• Many thanks to:
• Alexandra Olaya-Castro

(UCL)

• Tom Stace (UQ)
• Marshall Stoneham (UCL)
• Pawel Machnikowski (WUT)
• Howard Wiseman (GU)