3 from site 1 to site 2 since ω 1 31 ≈ ω 2 32 and C ( ω 1 $_{31}$) ≈ C ( ω 2 $_{32}$) (Fig. 1). TABLE I: Contributions of the system factor, ζ ab,ba , and of the overlap between transition frequency and phonon bath spectral properties, C ( ω ab ), to the calculation of the quantum kinetic rates k ab from energy eigenstate | e a 〉 to | e b 〉 . The two cases considered are described in Fig. 1. Case 1 Case 2 | e a 〉 → | e b 〉 log ζ log C log k log ζ log C log k 3 → 1 -1.5 11.6 10.0 -1.5 7.9 6.3 3 → 2 -2.4 10.7 8.3 -2.4 11.6 9.1 2 → 1 -4.3 11.4 7.1 -4.3 10.2 5.8 From the values of ζ and C ( ω$_{ab}$ ), it is clear that case 1 is designed such that an excitation starting on site 3 will tend to transfer to site 1, but in case 2 the population will go to site 2, albeit at different rates. Simulations of the quantum dynamics according to Redfield theory confirms this result (Fig. 2). Over the typical exciton lifetime of 1 ns in QDs, we not only achieve directed transfer, but also population enhancement compared to both the maximum site population during fully coherent dynamics and the population expected at thermal equilibrium. Moreover, while there are always experimental limi- tations in tuning parameters, the structure of Eq. 11 is valuable since it partitions the effects due to the system and bath. Using the calculated rates and/or a visual in- spection of C ( ω ) (Fig. 1B), it is easy to determine the impact of varying a system parameter on exciton trans- fer. Future work will address exciton and electron trans- fer between sites with varying spectral density functions, as well as the role of aligning dipole moment orienta- tions in engineering EET. We are also working to identify regimes in which preserved coherences enhance or reduce the efficiency of excitonic transfer. Of course, in situa- tions where multiple excitons are present in the system due to incident light intensity, frequency range, and/or optical spectral density of the quantum dots, the Hamil- tonian used to describe the system needs to be expanded accordingly [29]. To the extent that each exciton cou- ples to the environment through a spectral density as described in this letter, some of the intuition developed here should be transferable to these systems. However, there are a number of interesting complications, including many-body interactions among the excitons and correc- tions to the rates due to multi-phonon processes. A more careful analysis is needed when the offset in QD excita- tion energies is large compared with the characteristic frequency expanded by the phonon spectral densities or for systems at much higher temperature, e.g., room tem- perature, where multi-phonon processes are expected to be relevant, or even dominant. These effects are cur- rently under investigation and are beyond the scope of the current communication. In summary, we develop a framework for engineering environment-assisted and directed excitonic transfer in a network of coupled QDs based on a quantum kinetic rate approach. We emphasize the importance of how characteristic frequencies of a system fit within the spec- tral bath structure. Our examples utilize the factored and intuitive form of the population transfer rates equa- tion, which separates the contributions from the system (electronic) and bath (vibrational) degrees of freedom. This equation is similar in spirit to the rate equation for FRET, making it convenient to design interesting scenar- ios for environment-assisted transfer. Although we focus on QD examples, the principles presented here form the basis for engineering a wide range of desired EET in a variety of nanostructures or artificial molecular photo- synthetic units. The authors thank Semion Saikin for helpful discus- sions. A. A.-G. and A. P. were supported as part of the Center for Excitonics, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number de-sc0001088. L. V. acknowledges support from the NSF Graduate Research Fellowship. A.N. acknowl- edges support from NSERC (Canada) [1] G. D. Scholes, Annual Review of Physical Chemistry 54 , 57 (2003). [2] J. R. Lakowicz, Principles of Fluorescence Spectroscopy (New York: Springer, 2006), 3rd ed. [3] G. D. Scholes and G. Rumbles, Nat Mater 5 , 683 (2006). [4] S. A. Crooker, J. A. Hollingsworth, S. Tretiak, and V. I. Klimov, Physical Review Letters 89 , 186802 (2002). [5] E. Collini and G. D. Scholes, Science 323 , 369373 (2009). [6] H. Lee, Y. Cheng, and G. R. Fleming, Science 316 , 1462 (2007). [7] Y. Cheng and G. R. Fleming, Annual Review of Physical Chemistry 60 , 241 (2009). [8] M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay, and A. J. Landahl, Phys. Rev. A 71 , 032312 (2005). [9] E. Rozbicki and P. Machnikowski, Physical Review Let- ters 100 , 027401 (2008). [10] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, and A. Aspuru-Guzik, New Journal of Physics 11 , 033003 (2009). [11] M. B. Plenio and S. F. Huelga, New Journal of Physics 10 , 113019 (14pp) (2008). [12] A. A. High, E. E. Novitskaya, L. V. Butov, M. Hanson, and A. C. Gossard, Science 321 , 229 (2008). [13] K. M. Gaab and C. J. Bardeen, The Journal of Chemical Physics 121 , 7813 (2004). [14] J. Cao and R. J. Silbey, The Journal of Physical Chem- istry A 113 , 13825 (2009). [15] V. May and O. Kuhn, Charge and Energy Transfer Dy- namics in Molecular Systems (Wiley-VCH, 2004), 2nd ed.