Engineering directed excitonic energy transfer Alejandro Perdomo, Leslie Vogt, Ali Najmaie, and Alan Aspuru-Guzik Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, 02138, Cambridge, MA We provide an intuitive platform for engineering exciton transfer dynamics. We show that careful consideration of the spectral density, which describes the system-bath interaction, leads to opportu- nities to engineer the transfer of an exciton. Since excitons in nanostructures are proposed for use in quantum information processing and artificial photosynthetic designs, our approach paves the way for engineering a wide range of desired exciton dynamics. We carefully describe the validity of the model and use experimentally relevant material parameters to show counter-intuitive examples of a directed exciton transfer in a linear chain of quantum dots. The widely-applied F¨ orster theory for energy transfer links experimental results to estimates of system infor- mation, particularly in biological and nanoscale appli- cations [1, 2]. The usefulness of this theory is partly due to the simple expression of the kinetic rate constants as a product of electronic coupling and a spectral over- lap factor which captures the complexity of the environ- ment. F¨ orster theory describes transport in the inco- herent limit, but a complementary and more elaborate approach, such as Redfield theory, is often required to describe energy transfer. However, the information es- sential to understanding the dynamics is buried within the structure of the equations. In this letter, we employ a quantum kinetic rate approach to distill the informa- tion contained in equations into a simple, yet instructive, formula. We use this approach to design directed exciton transfer mediated by an environment. Excitonic energy transfer (EET) has been studied in systems as varied as quantum dot (QD) nanostructures [3, 4], polymer chains [5], and photosynthetic complexes [6, 7]. Many applications of EET would benefit from controlling exciton dynamics. Perfect state transfer, as studied in the quantum computing community, is achiev- able in certain engineered systems, but only at particular times during coherent evolution [8]. Recent works have shown that environment-induced decoherence can alter exciton dynamics [9–11], although controlling the trans- fer direction has only been achieved using external poten- tials [12]. Our paper builds upon the idea of engineering exciton transfer by designing appropriate system-bath in- teractions [13, 14]. We show that it is possible to design experimentally realizable systems where the environment can be used to direct the flow of energy. The Hamiltonian used in our simulation aims to cap- ture dynamics in a single-exciton manifold [15] interact- ing with an environment, with This representation is in the site basis {| s$_{n}$ 〉} of local- ized excitations on each of N sites, (e.g. QDs or chro- The equation of motion for the density operator ˆ ρ ( t ) in the excitonic energy basis representation, ˆ H$_{s}$ | e$_{a}$ 〉 = ϵ$_{a}$ | e$_{a}$ 〉 , is given by [17–19], To describe the excitonic quantum dynamics we use Redfield theory [17–19], which is a reduced density ma- trix approach in the regime of weak system-bath cou- pling. The formalism involves second-order perturbation theory in the system-bath interaction, ˆ H$_{sb}$ . This method assumes the Markov approximation, no initial correla- tions between system and bath degrees of freedom, and a thermalized bath. We avoid the frequently employed secular approximation (Bloch equations) [15, 19]; it is important to note that for systems where the time scale | ω$_{ab}$ − ω$_{cd}$ | − 1 is comparable or larger than the charac- teristic decoherence time, the coherence to population transfers contribute considerably to the dynamics of the system and must therefore be included. where g n q describes the site-specific coupling of electronic and vibrational degrees of freedom. We ignore the off- diagonal terms in the above equation, which correspond to phonon-induced modulations of the inter-site coupling, J$_{mn}$ [15]. While fluctuations of the gap give rise to the di- agonal electron-phonon coupling considered here [16], the inter-site couplings are usually one or two orders of mag- nitude smaller than the excitation gap and are therefore kept constant. In general, the validity of this approxima- tion is still an open question and its applicability varies from system to system. where b † q ( b$_{q}$ ) is the creation (destruction) operator for a phonon with wavevector q . The system-bath interaction is assumed to be linear, mophores), with excitation energy E$_{n}$ for each site and inter-site coupling J$_{mn}$ . The environment is described by a phonon bath, ˆ H$_{b}$ = ∑ q ℏ ω$_{q}$ ( b † $_{q}$b$_{q}$ + 1 / 2) , (3) ˆ H$_{sb}$ = N ∑ n =1 | s$_{n}$ 〉 〈 s$_{n}$ | ∑ q ℏ ω$_{q}$ ( g n $_{q}$b † q + ( g n $_{q}$) $^{∗}$b$_{q}$ ) . (4) ˆ H = ˆ H$_{s}$ + ˆ H$_{b}$ + ˆ H$_{sb}$ (1) ˆ H$_{s}$ = N ∑ n =1 E$_{n}$ | s$_{n}$ 〉 〈 s$_{n}$ | + ∑ n = m J$_{mn}$ | s$_{m}$ 〉 〈 s$_{n}$ | . (2) d ρ$_{ab}$ ( t ) d t = − iω$_{ab}$ρ$_{ab}$ ( t ) + ∑ cd R$_{ab,cd}$ρ$_{cd}$ ( t ) , (5) arXiv:1001.2602v1 [quant-ph] 15 Jan 2010