Theory of coherent resonance energy transfer

THE JOURNAL OF CHEMICAL PHYSICS 129, 101104 共2008兲 Theory of coherent resonance energy transfer Seogjoo Jang,1,a兲 Yuan-Chung Cheng,2 David R. Reichman,3 and Joel D. Eaves3 1 Department of Chemistry and Biochemistry, Queens College of the City University of New York, 65-30 Kissena Boulevard, Flushing, New York 11367, USA 2 Department of Chemistry, University of California, Berkeley, California 94720, USA 3 Department of Chemistry, Columbia University, 3000 Broadway, New York, New York 10027, USA 共Received 14 June 2008; accepted 13 August 2008; published online 11 September 2008兲 A theory of coherent resonance energy transfer is developed combining the polaron transformation and a time-local quantum master equation formulation, which is valid for arbitrary spectral densities including common modes. The theory contains inhomogeneous terms accounting for nonequilibrium initial preparation effects and elucidates how quantum coherence and nonequilibrium effects manifest themselves in the coherent energy transfer dynamics beyond the weak resonance coupling limit of the Förster and Dexter 共FD兲 theory. Numerical tests show that quantum coherence can cause significant changes in steady state donor/acceptor populations from those predicted by the FD theory and illustrate delicate cooperation of nonequilibrium and quantum coherence effects on the transient population dynamics. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2977974兴 The resonance energy transfer 共RET兲 of electronic excitations1 is an indispensable step in photosynthesis2 and organic optoelectronic processes.3 It also has a powerful spectroscopic application called fluorescence RET 共FRET兲, which can determine 2–10 nm distances in biological systems.4 How RET occurs is well understood at the level of the Förster and Dexter 共FD兲 theory,5–7 where the transfer rate can be calculated assuming incoherent quantum mechanical transitions. What happens if the transition falls in the coherent regime has become a topic of prime interest in recent years.8–11 The utilization of coherence may lead to highly efficient solar energy conversion devices12 and has significant implications in enhancing the sensitivity of FRET at short distances.13 However, the presence of coherence makes the definition of a transfer rate ambiguous, and assessing its effect on the overall RET dynamics has remained a difficult theoretical and experimental issue. The theory developed in this Communication elucidates some of these issues and provides a quantitative means to describe the RET dynamics in various limits. Let us consider the simplest system consisting of single chromophoric energy donor 共D兲 and acceptor 共A兲. The state where both D and A are in the ground electronic state is denoted as 兩g典. The state where only D 共A兲 is excited while A 共D兲 is in the ground electronic state is denoted as 兩D典 共兩A典兲. Only single electronic excitations are considered, and the three states constitute a complete set of system states. Initially, the system is in 兩g典, and the bath—all other degrees of freedom—is in equilibrium with 兩g典, with a corresponding Hamiltonian Hb. At time t = 0, a laser pulse with duration ␶pulse selectively excites 兩g典 to 兩D典. It is assumed that ␶pulse Ⰶ ␶RET, where the latter is the time scale of the RET dynamics. This in turn is a兲 Author to whom correspondence should be addressed. Electronic mail: seogjoo.jang@qc.cuny.edu. 0021-9606/2008/129共10兲/101104/4/$23.00 assumed to be much smaller than ␶sd, the spontaneous decay time to the ground state. Then, the RET dynamics for t ⬎ 0 共after the cessation of the pulse兲 can be described by a total Hamiltonian, H = Hsp + Hsc + Hsb + Hb, where Hsp and Hsc are system Hamiltonians 共with superscripts p and c representing population and coherence兲, and Hsb is the system-bath interaction Hamiltonian. These have the following forms: Hsp = ED兩D典具D兩 + EA兩A典具A兩, 共1兲 Hsc = J共兩D典具A兩 + 兩A典具D兩兲, 共2兲 Hsb = BD兩D典具D兩 + BA兩A典具A兩, 共3兲 where ED 共EA兲 is the energy of state 兩D典 共兩A典兲 relative to 兩g典, J is the resonance coupling between 兩D典 and 兩A典, and BD 共BA兲 represents the bath operator coupled to 兩D典 共兩A典兲. The total density operator at time t is denoted as ␳共t兲. The initial condition corresponding to the physical situation described above is ␳共0兲 = ␴共0兲e−␤Hb / Z, where ␤ = 1 / kBT, Z = Trb兵e−␤Hb其, and ␴共0兲 = 兩D典具D兩. For the Hamiltonians defined above, the corresponding quantum Liouville operators14 are denoted as L, Lsp, Lsc, Lsb, and Lb. Then, ␳共t兲 is governed by d␳共t兲 = − iL␳共t兲 = − i共Lsp + Lsc + Lsb + Lb兲␳共t兲. dt 共4兲 The major issue in coherent RET is that Lsp, Lsc, and Lsb are all comparable, which makes perturbation expansion in any of these unreliable. When the coupling to the bath is weak, the second order quantum master equation 共QME兲 approach15,16 may be employed, while for strong coupling to the bath, the FD theory5,6 is applicable. Our approach developed below interpolates between these limits by combining the polaron transformation17–19 and a QME formulation20 up to the second order. In order to make clear exposition of the theory, we here assume a spin-boson-type model.18,21 Thus, 129, 101104-1 © 2008 American Institute of Physics Downloaded 22 Jan 2011 to 140.112.115.121. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 101104-2 Jang et al. J. Chem. Phys. 129, 101104 共2008兲 Hb = 兺nប␻n共b†nbn + 21 兲, where b†n 共bn兲 is the creation 共annihilation兲 operator of the nth mode with frequency ␻n, and BD = 兺nប␻ngnD共bn + b†n兲 and BA = 兺nប␻ngnA共bn + b†n兲. It is assumed that J in Eq. 共2兲 is a time independent parameter. Numerous theoretical studies have been made for this model, but its dynamics in the ranges of parameters corresponding to coherent RET remains relatively unknown.11 Applying the polaron transformation17–19 generated by G = 兺n共b†n − bn兲共gnD兩D典具D兩 + gnA兩A典具A兩兲 to Eq. 共4兲, we obtain the following time evolution equation for ˜␳共t兲 = eG␳共t兲e−G: d˜␳共t兲 = − i共L̃sp + L̃sc + Lb兲˜␳共t兲, dt 共5兲 where L̃sp and L̃sc are quantum Liouville operators for H̃sp = ẼD兩D典具D兩 + ẼA兩A典具A兩, 共6兲 2 and ẼA = EA − 兺ngnA ប␻n. In † † −兺ngnA共bn−bn兲 and , and ␪D ␪A = e In Eq. 共6兲, Eq. 共7兲, ␪D = e and ␪A† are their Hermitian conjugates. Accordingly, the initial † −␤Hb condition transforms to ˜␳共0兲 = ␴共0兲␪D ␪D / Z, which is e nonequilibrium with respect to the bath.22 For the purpose of deriving the QME, we divide the total transformed Hamiltonian as H̃ = H̃0 + H̃1. The zeroth order term H̃0 is defined as H̃0 = H̃sp + 具H̃sc典 + Hb = H̃0,s + Hb , 共8兲 / Zb, and H̃0,s = ẼD兩D典具D兩 + ẼA兩A典具A兩 + Jw共兩D典具A兩 + 兩A典具D兩兲. 共9兲 2 −兺ncoth共␤ប␻n/2兲␦gn/2 † with ␦gn = gnD ␪A典 = 具␪A† ␪D典 = e Here, w = 具␪D − gnA. The remaining first order term H̃1 is defined as H̃1 = H̃sc − 具H̃sc典 = J共B̃兩D典具A兩 + B̃†兩A典具D兩兲, 共10兲 † where B̃ = ␪D ␪A − w. By definition, 具B̃典 = 具B̃†典 = 0. A crucial point to note is that, unlike the usual assumption of the FD theory,5,6 we follow the approach of Abram and Silbey23 and take JB̃ and JB̃† as perturbations which remain small in both limits of weak and strong system-bath couplings. This allows for the second order QME with respect to H̃1 to be valid in both limits. ˜ In the interaction picture of H̃0, ˜␳I共t兲 = eiL0t˜␳共t兲 is governed by the following time evolution equation: d ˜␳I共t兲 = − iL̃1,I共t兲˜␳I共t兲, dt 共11兲 t d␶PL̃1,I共t兲L̃1,I共␶兲 0 共13兲 where PL̃1,I共t兲P = 0 has been used and Q˜␳共0兲 = ␴共0兲 † −␤Hb e ␪D − e−␤Hb兲 / Z. In Eq. 共13兲, P˜␳I共␶兲 can be replaced ⫻共␪D with P˜␳I共t兲 without affecting the accuracy up to the second order.20 Taking the trace of the resulting equation over the bath degrees of freedom, we obtain the following time-local QME for ˜␴I共t兲 = Trb兵˜␳I共t兲其: d ˜␴I共t兲 = − R共t兲˜␴I共t兲 + I共t兲, dt 共14兲 R共t兲 = 冕 t d␶ Trb兵L̃1,I共t兲L̃1,I共␶兲␳b其, 共15兲 0 I共t兲 = − i Trb兵L̃1,I共t兲Q˜␳共0兲其 − 冕 t d␶ Trb兵L̃1,I共t兲QL̃1,I共␶兲˜␳共0兲其. 共16兲 0 While being time local, Eq. 共14兲 can account for nonMarkovian bath effects through the time dependence of R共t兲 and is expected to show good performance beyond the typical perturbative regime, as has been demonstrated for other cases.24 As long as w defined below Eq. 共9兲 is nonzero, it is simple to show that Eq. 共14兲 captures the Redfield limit15 for long times and weak system-bath coupling limit. Two straightforward extensions of Eq. 共14兲 are possible. The first is for more general initial condition with coherent mixture of 兩D典 and 兩A典, which is important for modeling pump-probe spectroscopy. The second is multistate generalization. Inserting Eq. 共12兲 into Eqs. 共15兲 and 共16兲 and using the cyclic invariance of the bath operators within Trb兵¯其, we can explicitly decouple the bath correlation functions from the commutators of system operators. The resulting expression for Eq. 共15兲, when applied to ˜␴I共t兲, can be shown to be R共t兲˜␴I共t兲 = J2 −K共0兲 e ប2 + 共e 冕 K共t−␶兲 t d␶兵共e−K共t−␶兲 − 1兲关T共t兲,T共␶兲˜␴I共t兲兴 0 − 1兲关T†共t兲,T共␶兲˜␴I共t兲兴 + 共eK共t−␶兲 − 1兲关T共t兲,T†共␶兲˜␴I共t兲兴 + 共e−K共t−␶兲 − 1兲关T†共t兲,T†共␶兲˜␴I共t兲兴其 + H . c . , 共17兲 where L̃1,I共t兲 is the quantum Liouville operator for H̃1,I共t兲 = J共B̃共t兲T共t兲 + B̃†共t兲T†共t兲兲, 共12兲 ˜ 冕 ⫻共Q˜␳共0兲 + P˜␳I共␶兲兲, 共7兲 2 ẼD = ED − 兺ngnD ប␻n † −兺ngnD共bn−bn兲 where 具¯典 denotes average over e d P˜␳I共t兲 = − iPL̃1,I共t兲Q˜␳共0兲 − dt where † ␪A兩D典具A兩 + ␪A† ␪D兩A典具D兩兲. H̃sc = J共␪D −␤Hb both the homogeneous and inhomogeneous terms consistently, we obtain ˜ with B̃共t兲 = eiHbt/បB̃e−iHbt/ប and T共t兲 = eiH0,st/ប兩D典具A兩e−iH0,st/ប. Applying the standard projection operator technique20 with P共·兲 ⬅ ␳b Trb兵·其 and Q = 1 − P to Eq. 共11兲 and making second order approximations 关with respect to L̃1,I共t兲兴 for K共t兲 = 兺n␦g2n兵coth共␤ប␻n / 2兲cos共␻nt兲 − i sin共␻nt兲其 and where “H.c.” represents the Hermitian conjugates of all the previous terms. The same convention will be used hereafter. The expression for Eq. 共16兲 is more complicated because it involves nonequilibrium bath correlation functions. After careful examination, we find that it can be expressed compactly in terms of K共t兲, a new bath function f共t兲 = e2i兺ngnD␦gn sin共␻nt兲, Downloaded 22 Jan 2011 to 140.112.115.121. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 101104-3 Coherent resonance energy transfer J. Chem. Phys. 129, 101104 共2008兲 1 which represents the correlation of the initial donor bath, and f m共t , ␶兲 = f共t兲f共␶兲 − 1 and f a共t , ␶兲 = f共t兲 + f共␶兲 − 2. Thus, Eq. 共16兲 can be shown to be iJ −K共0兲/2 e 共f共t兲 − 1兲关T共t兲, ␴共0兲兴 ប J2 − 2 e−K共0兲 ប 冕 0.4 0.4 0.2 (a) ∆ E=1, J=0.5 PD(t) I共t兲 = − 0.6 1 w/o I(t) Full 0.8 r PD (t) KD(t) 0.6 0.8 t d␶兵F共1兲共t, ␶兲关T共t兲,T共␶兲␴共0兲兴 0 + F共2兲共t, ␶兲关T†共t兲,T共␶兲␴共0兲兴 † + F共3兲共t, ␶兲关T共t兲,T 共␶兲␴共0兲兴 0 0 1 共18兲 where F共1兲共t , ␶兲 = f m共t , ␶兲e−K共t−␶兲 − f a共t , ␶兲, F共2兲共t , ␶兲 = f m K共t−␶兲 − f a共−t , ␶兲, F共3兲共t , ␶兲 = f m共t , −␶兲eK共t−␶兲 − f a共t , −␶兲, 共−t , ␶兲e and F共4兲共t , ␶兲 = f m共−t , −␶兲e−K共t−␶兲 − f a共−t , −␶兲. In the above expressions, the system-bath coupling is fully specified by two spectral densities, Js共␻兲 = 兺n␦共␻ − ␻n兲␻2n␦g2n and Ji共␻兲 = 兺n␦共␻ − ␻n兲␻2ngnD␦gn. These spectral densities can represent various situations including the cases where there are common bath modes25 共gnDgnA ⫽ 0兲 between the donor and the acceptor. It is noteworthy to mention important qualitative features related to the characteristics of the spectral density. Let us assume that Js共␻兲 ⬀ ␻ p in the limit of ␻ → 0. If p ⱕ 1, e−K共0兲 = 0 and 具H̃s1典 = 0 at all temperatures. Then, the only surviving terms are those with e−K共0兲+K共t−␶兲 in Eq. 共17兲 and those with F共2兲共t , ␶兲 or F共3兲共t , ␶兲 in Eq. 共18兲. The resulting dynamics involves only population terms 共兩D典具D兩 and 兩A典具A兩兲 in a way similar to the noninteracting blip approximation,21 but our equations are time local and include inhomogeneous terms. On the other hand, for p ⬎ 2, 具H̃s1典 and e−K共0兲 are nonzero at all temperatures, and the dynamics always involves coherence terms, 兩D典具A兩 and 兩A典具D兩. However, caution is required, and it is important to identify the physical origin of the low frequency modes especially for the Ohmic density. If the low frequency modes have an anharmonic origin, they may not make full multiphonon contributions or remain virtually static during the lifetime of the electronic excitation. This situation can be accounted for by introducing a lower bound of order 1 / ␶sd in the frequency-domain integration of the spectral density. With this modification, our theory reduces to the Redfield approach for weak coupling limit even for Ohmic spectral densities. For numerical calculations, it is convenient to express Eq. 共14兲 in the eigenbasis of H̃0,s. Detailed expressions are provided in the supporting document. Numerical tests have been made for the following super-Ohmic spectral densities: Js共␻兲 / 2 = Ji共␻兲 = 共␩ / 3!兲共␻3 / ␻2c 兲e−␻/␻c. In the units where ␻c = ប = 1 and kBT = 1, two different cases of ␩ = 1 and 3 were considered for ⌬E = ẼD − ẼA = ⫾ 1. Calculations of ˜␴I共t兲 have been made both with and without I共t兲 in Eq. 共14兲, which can then be used to determine any physical observable of the system. Here, we focus only on the population of the excited ˜ ˜ donor, PD共t兲 = 具D兩e−iH0,st/ប˜␴I共t兲eiH0,st/ប兩D典. In order to assess r 共t兲 based the role of quantum coherence, the population PD on the following rate equation was also calculated: 10 15 0.8 0.8 0.6 0.6 0.4 0.4 0.2 (b) ∆E=−1, J=0.5 0.2 (d) ∆E=−1, J=2 0 + F共4兲共t, ␶兲关T†共t兲,T†共␶兲␴共0兲兴其 + H . c . , 5 0.2 (c) ∆E=1, J=2 0 20 0 2 1 0 5 10 15 20 0 0 2 4 6 4 6 Time FIG. 1. 共Color online兲 Time-dependent donor populations for ␩ = 1. Units are such that ប = ␻c = 1 and kBT = 1. Blue dashed lines 关without I共t兲兴 result from Eq. 共14兲 with the inhomogeneous term I共t兲 = 0, and black solid lines 共full兲 correspond to the results of Eq. 共14兲 employing the full expressions. Red dot-dashed lines represent results based on Eq. 共19兲. Red dotted lines correspond to the time-dependent equilibrium donor population KD共t兲 as defined below Eq. 共19兲. Different panels show results for different values of ⌬E = ED − EA and J as indicated. d r r r r r P 共t兲 = − kDA 共t兲PD 共t兲 + kAD 共t兲共1 − PD 共t兲兲, dt D 共19兲 r where kDA 共t兲 is the time-dependent FD rate7 from D to A given by r 共t兲 = kDA 冋冕 2J2 −K共0兲 e Re ប2 t 0 ˜ ˜ 册 d␶ei共ED−EA兲t/ប共eK共t兲 − 1兲 . 共20兲 r The expression for kAD 共t兲 is the same except for the replacement ẼD − ẼA → ẼA − ẼD. The time-dependent equilibrium r r r constant of the donor, KD共t兲 = kAD 共t兲 / 共kDA 共t兲 + kAD 共t兲兲, was also calculated as a reference. Figure 1 shows results for ␩ = 1. When J = 0.5, the quantum coherence causes oscillatory donor population, but its average over the period and the steady state limit remain very close to those based on the rate equation, Eq. 共19兲. When J = 2, the quantum coherence has significant effects on the steady state donor population. For ⌬E = 1, there is more donor population 共less efficient transfer兲 than the prediction of the rate equation, Eq. 共19兲. For ⌬E = −1, the opposite is true. Thus, quantum coherence 共or tunneling兲 is shown to counteract the prescription of the detailed balance based on the FD rate equation 共in the site localized basis兲. Figure 2 shows results for ␩ = 3. When J = 0.5, the system-bath coupling is large enough to damp the oscillatory population and to make the time-dependent population nearly overlap with that based on the rate equation. When J = 2, there are slight transient oscillations at early times, and the steady state donor populations differ from those of the rate equation but to less extents than those in Fig. 1. Also shown are significant contributions of the inhomogeneous term I共t兲 on the transient behavior of population dynamics. This suggests the importance of including nonequilibrium effects for quantitative description of the ultrafast RET dynamics. For the modeling of nonlinear spectroscopy experiments being used to probe such dynamics in real time, fur- Downloaded 22 Jan 2011 to 140.112.115.121. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 101104-4 Jang et al. J. Chem. Phys. 129, 101104 共2008兲 1 0.6 1 w/o I(t) Full 0.8 r PD (t) KD(t) 0.6 0.4 0.4 0.8 PD(t) 0.2 (a) ∆ E=1, J=0.5 0 0 1 5 10 15 3 0.2 (c) ∆E=1, J=2 0 20 0 2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 (b) ∆E=−1, J=0.5 0.2 (d) ∆E=−1, J=2 0 0 5 10 15 20 0 0 2 4 6 4 6 Time FIG. 2. 共Color online兲 Time-dependent donor populations for ␩ = 3. All other details are the same as in Fig. 1. ther extension of the theory for more general initial conditions as was done by Matro and Cina16 is needed, which is possible at the expense of more complicated I共t兲 in our formalism. Future theoretical efforts will be dedicated to this issue. In summary, we have developed a theory of coherent RET including both nonequilibrium and quantum coherence effects. Numerical tests demonstrate the presence of oscillatory population dynamics even for moderately large systembath coupling and interesting effects of quantum coherence on the steady state donor populations. ACKNOWLEDGMENTS We thank Bob Silbey and Graham Fleming for discussions. Support for S.J. comes from CUNY-collaborative Incentive Grant No. 80209-10-13 and ACS-PRF Grant No. 46735-G6. D.R.R. and J.D.E. thank the support of the NSF with Grant No. CHE-0719089. 1 B. J. Schwartz, Annu. Rev. Phys. Chem. 54, 141 共2003兲. S. Weiss, Science 283, 1676 共1999兲. 5 Th. Förster, Discuss. Faraday Soc. 27, 7 共1959兲. 6 D. L. Dexter, J. Chem. Phys. 21, 836 共1953兲. 7 S. Jang, Y. J. Jung, and R. J. Silbey, Chem. Phys. 275, 319 共2002兲; S. Jang, M. D. Newton, and R. J. Silbey, Phys. Rev. Lett. 92, 218301 共2004兲; S. Jang, J. Chem. Phys. 127, 174710 共2007兲. 8 G. S. Engel, T. R. Calhoun, E. L. 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