7 FIG. 6: Time evolution of the exciton probability | ψ$_{n}$ ( t ) | 2 in real space using ˜ D Ansatz. The unit of time is 1 /ω 0 . (a) J = 0 . 1 , W = 0 . 1 , S = 0 . 5; (b) J = 0 . 5 , W = 0 . 8 , S = 0 . 5; (c) J = 0 . 5 , W = 0 . 1 , S = 0 . 5; (d) J = 1 . 0 , W = 0 . 8 , S = 0 . 5; (e) J = 1 . 0 , W = 0 . 8 , S = 2 . 0. FIG. 7: Time evolution of the phonon displacements | λ$_{n}$ ( t ) | of the ˜ D Ansatz in real space. The un it of time is 1 /ω 0 . (a) J = 0 . 1 , W = 0 . 1 , S = 0 . 5; (b) J = 0 . 5 , W = 0 . 8 , S = 0 . 5; (c) J = 0 . 5 , W = 0 . 1 , S = 0 . 5; (d) J = 1 . 0 , W = 0 . 8 , S = 0 . 5; (e) J = 1 . 0 , W = 0 . 8 , S = 2 . 0. placements λ$_{n}$ ( t ) and β$_{n}$ ( t ). Fig. 7c shows that when W is small, most of λ$_{n}$ ( t ) will be localized on the n = 0 site. As the phonon band width incr eases, λ$_{n}$ ( t ) will form two traveling wave packets with op posite directions and propagate to the entire chain as sho wn in Fig. 7b. The dynamics of β$_{n}$ ( t ) is more c omplex. As shown in Figs. 7b and 7c, the phonon displac ements described by parameter λ$_{n}$ ( t ) propagate from t he n = 0 site to the rest of the aggregate. But the same is not true for β$_{n}$ ( t ). Most of β$_{n}$ ( t ) will stay localized o n the site of exciton of the localized phonon wave packets related to β$_{n}$ ( t ) be- comes faster J is increased from 0 . 5 to 1 . 0 as shown in Fig. 8. As for the ˜ D Ansatz, an increase in the width of the phonon dispersion W will substantially deepen the phonon dissipative effect on the exciton amplitude ψ$_{n}$ ( t ), and as a result the exciton coherence quickly vanishes from the case of W = 0 . 8. (cf. Fig. 6b and 6c). W is also one main factor which determines the velocity of the phonon wave packets for both types of phonon dis-