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. T c 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) Incoherent transfer Weak exciton-phonon coupling

Motivation • Energy transfer is ubiquitous • Experimental observation of QDs: B. D. Gerardot et al., Phys. Rev. Lett. 95, 137403 (2005) coherent effects • Conditions for coherent or incoherent transfer? Conjugated Polymers: E. Collini and G. D. Scholes, Science 323, 369 (2009) • How can coherence survive at elevated temperatures? LH1-RC: Fig. courtesy of A. Olaya-Castro • Can we understand observed dynamics from simple models? 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 Donor Acceptor g 1 g 2 • Weak - strong system bath | X � | X � V ǫ 2 ǫ 1 coupling (single to multi- | 0 � | 0 � phonon effects) d Environment • Low - high temperatures � ω k b † H = ǫ 1 | X � 1 � X | + ǫ 2 | X � 2 � X | + V ( | 0 X �� X 0 | + | X 0 �� 0 X | ) + k b k • Correlated - uncorrelated k � ( g k, 1 b † � ( g k, 2 b † dephasing fluctuations + | X � 1 � X | k + g ∗ k, 1 b k ) + | X � 2 � X | k + g ∗ k, b k ) k k • Polaron transformation g k, 1 = | g k | e i k · d / 2 g k, 2 = | g k | e − i k · d / 2

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

Previous works (II) Modified Redfield treatment Non-Markovian dynamics Polaron transformation: Interpolates weak to strong system bath interactions

Model Hamiltonian • Single excitation subspace, map to a 2-level system Effective 2LS ω k b † � g 1 − g 2 H sub = ǫ 1 | 0 �� 0 | + ǫ 2 | 1 �� 1 | + V ( | 1 �� 0 | + | 0 �� 1 | ) + k b k | 1 � k | 0 X � � � � ( g k, 1 − g k, 2 ) b † +( | 0 �� 0 | − | 1 �� 1 | ) k + ( g k, 1 − g k, 2 ) ∗ b k V | 0 � | X 0 � k • Polaron transformation Environment e ± s = | X 0 �� X 0 | Π k D ( g k, 1 / ω k ) + | 0 X �� 0 X | Π k D ( g k, 2 / ω k ) Effective spectral density H P = e s He − s = ǫ ω k b † � 2 σ z + V R σ x + k b k + V ( σ x B x + σ y B y ) F 1 D ( ω , d ) = cos ( ω d/c ) J � Ω �� 1 � F � Ω ,d �� � ps � 1 � k 8 F 2 D ( ω , d ) = J 0 ( ω d/c ) 6 F 3 D ( ω , d ) = sinc( ω d/c ) 4 • Bath-renormalised interaction 2 R ∞ d ω J ( ω ) ω 2 (1 − F ( ω ,d )) coth ω / 2 k B T V 0 V R = BV = e − 0 0 5 10 15 20 Ω � ps � J ( ω ) = αω 3 e − ( ω / ω c ) 2

Master equation • Treat fluctuations to second order: Born-Markov approximation 2[ σ z , ρ ] − V 2 ρ = − i η � (( γ l ( ω ′ ) + 2 iS l ( ω ′ ))[ P l ( ω ) , P l ( ω ′ ) ρ ] + H.c. ) ˙ 2 l, ω , ω ′ � ω , ω ′ ∈ { 0 , ± η } ǫ 2 + 4 V 2 η = R 0.5 • Rates are FTs of bath correlation functions � � 0 0.0 0 20 � ∞ 10 γ x/y ( ω ) = e ω /k B T d τ e i ωτ Λ x/y ( τ ) 5 0 x Τ � � 10 −∞ 10 � 20 � ∞ d ω J ( ω ) cos ωτ • In terms of phonon propagator ϕ ( τ ) = 2 ¯ ω 2 (1 − F ( ω , d )) sinh ( ω / 2 k B T ) 0 Λ x ( τ ) = B 2 Λ y ( τ ) = B 2 ϕ ( τ ) + e − ¯ 2 ( e ¯ ϕ ( τ ) − 2) , 2 ( e ¯ ϕ ( τ ) − e − ¯ ϕ ( τ ) )

Resonant dynamics • Initialise in donor, look at � � 2 + ( Γ 2 − Γ 1 ) cos ξ t sin ξ t � σ z � t = e − ( Γ 1 + Γ 2 ) t/ 2 subsequent excitation 2 ξ dynamics � � 2 γ x (0) + γ y (2 V R )(1 + 2 N (2 V R )) ǫ = 0 Γ 1 = V 2 Γ 2 = 2 V 2 γ x (0) , , (1 + N (2 V R )) N ( ω ) = ( e ω /k B T − 1) − 1 • Coherent or incoherent � 16 V 2 R − ( Γ 1 − Γ 2 ) 2 ξ ≈ dynamics possible 1.0 1.0 1.0 0.5 0.5 0.5 �Σ z � t �Σ z � t �Σ z � t 0.0 0.0 0.0 � 0.5 � 0.5 � 0.5 � 1.0 � 1.0 � 1.0 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 t t t 4 V R > ( Γ 1 − Γ 2 ) → ξ real 4 V R = ( Γ 1 − Γ 2 ) → ξ = 0 4 V R < ( Γ 1 − Γ 2 ) → ξ imaginary Coherent Crossover Incoherent

Weak coupling - coherent 1.0 • Expand bath correlation functions to first order in to get single-phonon rates 0.5 ϕ ( τ ) ¯ �Σ z � t 0.0 B 2 ¯ Λ y ( τ ) ≈ ˜ Λ x ( τ ) ≈ 0 , ϕ ( τ ) � 0.5 � 1.0 0 50 100 150 200 • Damped oscillations t � σ z � t = e − ˜ Γ 1 t/ 2 [cos (˜ ξ t/ 2) − (˜ Γ 1 / ˜ ξ ) sin (˜ ξ t/ 2)] 0.04 1D 2D 0.03 � • is real ˜ 16 ˜ R − ˜ V 2 Γ 2 ξ = 1 � ps � 1 � 1 0.02 3D � � 0.01 • Damping consistent with a weak- 0.00 coupling treatment 0 5 10 15 20 25 d � nm � Γ 1 = π J (2 ˜ ˜ V R )(1 − F (2 ˜ V R , d )) coth ( ˜ V R /k B T )

When are single phonon rates valid? • Example: consider a particular spectral density 0.5 � J ( ω ) = αω 3 0.0 � 0 0 20 10 • This leads to: 5 0 x Τ � � 10 10 � 20 � sech 2 τ ′ − tanh ( x − τ ′ ) + tanh ( x + τ ′ ) � ϕ ( τ ′ ) = ϕ 0 ¯ 0.0020 � � 1 �� 2 �� 2 � arb. units � 2 x 0.0015 T d small • Two important temperature scales: 0.0010 0.0005 B T 2 = T 2 /T 2 ϕ 0 = 2 π 2 α k 2 0 , 0.0000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x = π k B Td/c = T/T d T � T 0 • Weak fluctuation correlations 0.0005 � � 1 �� 2 �� 2 � arb. units � 0.0004 T d large x ≫ 1( T ≫ T d ) → ϕ 0 ≪ 1 0.0003 • Strong fluctuation correlations 0.0002 0.0001 x ≪ 1( T ≪ T d ) → ϕ 0 x 2 ≪ 1 0.0000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T � T 0

Incoherent - High temperature, multi-phonon 1.0 • Estimate rates by saddle-point 0.5 approximation (essentially �Σ z � t expand about ) 0.0 ϕ ( τ ) ¯ τ = 0 � 0.5 0 e 2 ϕ 0 / 3 e ϕ 0 (2 x csch2 x − 1) /x 2 Γ 1 ≈ 2 Γ 2 ≈ 2 β V 2 F B 2 � 1.0 0 50 100 150 200 � πϕ 0 ( x − sech 2 x tanh x ) /x t • For high enough temperature 10 we have incoherent transfer saddle point � � 1 �� 2 �� 2 � arb. units � 1 � 0.1 1 / 4 ≈ i Γ 1 / 2 → � σ z � t ≈ e − Γ 1 t 16 V 2 R − Γ 2 ξ ≈ full 0.01 0.001 weak • Considering multi-phonon 10 � 4 processes gives rise to a 0 1 2 3 4 5 6 7 T � T 0 destruction of coherent effects

Crossover • Crossover generally occurs in 4 the high temperature regime. 4 V R 3 2 • Simplified condition: 8 V R = Γ 1 1 Γ 1 − Γ 2 8 V R > Γ 1 → coherent 0 8 V R < Γ 1 → incoherent 0 2 4 6 8 T � T 0 Strong correlation • Critical temperature T c 10 8 weak V F B 0 e 5 ϕ c / 6 e ϕ c (coth x c − 2 tanh x c − 1 /x c ) / 2 x c T 2 c = T 0 , 6 correlation T c � T 0 � π ( x c − sech 2 x c tanh x c ) /x c 4 k B 4 2 ϕ c = T 2 c /T 2 x c = T c /T d 0 , 0 0 1 2 3 4 • Strong correlations suppress d � d 0 � T 0 � T d multi-phonon effects

Far off-resonance • Pairs of QDs not usually naturally resonant V/ ǫ ≪ 1 QDs: B. D. Gerardot et al., Phys. Rev. Lett. 95, 137403 (2005) • Expand to 2nd order in V/ ǫ � σ z � t ≈ e − Γ t − (1 − e − Γ t ) tanh ( βǫ / 2) • Incoherent transfer from donor to acceptor, as expected Γ = V 2 (1 + 2 N ( ǫ )) (1 + N ( ǫ )) ( γ x ( ǫ ) + γ y ( ǫ )) • Weak coupling (single-phonon): Γ ≈ (4 π ˜ ˜ V 2 R / ǫ 2 ) J ( ǫ )(1 − F ( ǫ , d )) coth ( βǫ / 2) comparison with Rozbicki and Machnikoski 0.01 � Inc � arb. units � 0.001 10 � 4 • High-temperature: � σ z � t ≈ e − Γ t 10 � 5 10 � 6 0.0 0.5 1.0 1.5 2.0 2.5 T � T 0

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. T c 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))

Acknowledgements • Many thanks to: • Funding • Alexandra Olaya-Castro (UCL) • Tom Stace (UQ) • Marshall Stoneham (UCL) • Pawel Machnikowski (WUT) • Howard Wiseman (GU)