13 Appendix A: Third Order Response Functions In this appendix we list the third order response func- tion used in calculating the impulsive 2D spectra. The first index of the response function follows the standard notation of Ref. [3]. The second index is g for pathways not involving the two-exciton band, while all pathways denoted by f include a two-exciton contribution (see e.g. Ref. [11]). R$_{3}$$_{g}$ ( t, T, τ ) = tr { µ$_{ge}$ U$_{egeg}$ ( t ) V ( L ) eg × U$_{eeee}$ ( T ) V ( L ) eg U$_{gege}$ ( τ ) V ( R ) ge ρ$_{0}$ } , (A2) R$_{2}$$_{g}$ ( t, T, τ ) = tr { µ$_{ge}$ U$_{egeg}$ ( t ) V ( R ) eg × U$_{eeee}$ ( T ) V ( R ) ge U$_{egeg}$ ( τ ) V ( L ) eg ρ$_{0}$ } , (A1) R$_{1}$$_{g}$ ( t, T, τ ) = tr { µ$_{ge}$ U$_{egeg}$ ( t ) V ( R ) eg × U$_{eeee}$ ( T ) V ( L ) eg U$_{gege}$ ( τ ) V ( R ) ge ρ$_{0}$ } . (A6) R$_{2}$$_{f}$ ( t, T, τ ) = tr { µ$_{fe}$ U$_{efef}$ ( t ) V ( R ) ef R$_{1}$$_{f}$ ( t, T, τ ) = tr { µ$_{fe}$ U$_{efef}$ ( t ) V ( R ) ef × U$_{eeee}$ ( T ) V ( R ) ge U$_{egeg}$ ( τ ) V ( L ) eg ρ$_{0}$ } (A5) × U$_{gggg}$ ( T ) V ( L ) ge U$_{egeg}$ ( τ ) V ( L ) eg ρ$_{0}$ } , (A4) R$_{4}$$_{g}$ ( t, T, τ ) = tr { µ$_{ge}$ U$_{egeg}$ ( t ) V ( L ) eg × U$_{gggg}$ ( T ) V ( R ) eg U$_{gege}$ ( τ ) V ( R ) ge ρ$_{0}$ } , (A3) Operators and superoperators used in this appendix are defined in Section IV A. [34] T. Mančal, L. Valkunas, and G. R. Fleming, Spec- [33] A. Ishizaki and G. R. Fleming, J. Chem. Phys. 130 , 243111 (2009). [32] A. Ishizaki and G. R. Fleming, J. Chem. Phys. 130 , 243110 (2009). [31] S. J. Jang, M. D. Newton, and R. J. Silbey, Phys. Rev. Lett. 92 , 218301 (2004). [30] M. N. Yang and G. R. Fleming, Chem. Phys. 275 , 355 (2002). [29] W. M. Zhang, T. Meier, V. Chernyak, and S. Mukamel, J. Chem. Phys. 108 , 7763 (1998). [28] V. Čápek, I. Barvík, and P. Heřman, J. Luminiscence 204 , 306 (2004). [27] V. May and O. Kühn, Charge and Energy Transfer Dy- namics in Molecular Systems (Kluwer Academic Publish- ers, Dordrecht, 2000). [26] S. Mukamel, Chem. Phys. 37 , 33 (1979). [25] S. Mukamel, I. Oppenheim, and J. Ross, Phys. Rev. A 17 , 1988 (1978). [24] F. Shibata, Y. Takahashi, and N. Hashitsume, J. Stat. Phys. 17 , 171 (1977). [23] N. Hashitsume, F. Shibata, and M. Shingu, J. Stat. Phys. 17 , 155 (1977). [22] R. Zwanzig, Physica 30 , 1109 (1964). [21] S. Nakajima, Progr. Theor. Phys. 20 , 948 (1958). [20] A. Olaya-Castro, C. F. Lee, F. F. Olsen, and N. F. John- son, Phys. Rev. B 78 , 085115 (2008). [19] M. Plenio and S. F. Huelga, New J. Phys. 10 , 113019 (2008). [18] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru- Guzik, J. Chem. Phys. 129 , 174106 (2008). [17] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, and A. Aspuru-Guzik, New J. Phys. 11 , 033003 (2009). [16] E. Collini and G. D. Scholes, Science 323 , 369 (2009). [15] H. Lee, Y.-C. Cheng, and G. R. Fleming, Science 316 , 1462 (2007). [14] G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, Nature 446 , 782 (2007). [13] P. Kjellberg, B. Brüggemann, and T. Pullerits, Phys. Rev. B 74 , 024303 (2006). [12] A. V. Pisliakov, T. Mančal, and G. R. Fleming, J. Chem. Phys. 124 , 234505 (2004). [11] T. Brixner, T. Mančal, I. V. Stiopkin, and G. R. Fleming, J. Chem. Phys. 121 , 4221 (2004). [10] T. Brixner, I. V. Stiopkin, and G. R. Fleming, Opt. Lett. 29 , 884 (2004). [9] M. L. Cowan, J. P. Ogilvie, and R. J. D. Miller, Chem. Phys. Lett. 386 , 184 (2004). [8] D. M. Jonas, Annu Rev. Phys. Chem 54 , 425 (2003). [7] A. Damjanovic, I. Kosztin, U. Kleinekathöfer, and K. Schulten, Phys. Rev. E 65 , 031919 (2002). [6] M. E. Madjet, A. Abdurahman, and T. Renger, J. Phys. Chem. B 110 , 17268 (2006). [5] B. P. Krueger, G. D. Scholes, and G. R. Fleming, J. Phys. Chem. B 102 , 5378 (1998). [4] T. Renger and R. A. Marcus, J. Chem. Phys. 116 , 9997 (2002). [3] S. Mukamel, Principles of nonlinear spectroscopy (Oxford University Press, Oxford, 1995). [2] H. van Amerongen, R. van Grondelle, and L. Valkunas, Photosynthetic Excitons (Kluwer Academic Publishers, Dordrecht, 2000). [1] Quantum Dynamics of Complex Molecular Systems , edited by D. A. Micha and I. Burghardt (Springer, Berlin, 2007).