6 tional to the transfer integral J . The Huang-Rhys factor S is set at 0 . 5. In Figs. 3 and 4 ( J = 0 . 5), the first trav- eling wave of the exciton amplitude reaches the the op- posite end of the ring ( n = 16) at t ≈ 2 . 6(2 π/ω$_{0}$ ), while in Fig. 5 ( J = 1 . 0), the exciton amplitude reaches the n = 16 site at t ≈ 1 . 3(2 π/ω$_{0}$ ). This is because the prop- agation speed of the exciton, or equivalently, ∂E$_{ex}$ ( k ) /∂k with E$_{ex}$ ( k ) the bare exciton band, is proportional to the exciton transfer integral J according to Eq. (2). The ef- fect of phonon dispersion on the exciton is reflected in the differing degree of phonon-induced dissipation in Figs. 3b and 4b. Exciton wave packets in Fig. 4b survive much longer than those in Fig. 3b thanks to a much smaller phonon band width. At t = 0 there are no phonon deformations anywhere on the one-dimensional ring. Due to photo-excitation at t = 0, a high concentration of the exciton density at a single location, appears in the vicinity of n = 0 for a short time duration near t = 0 [the red-colored spot near the point (0 , 0) in Figs. 3b, 4b and 5b], triggering a pair of localized phonon wave packets which travel at group velocities ± v$_{q}$ with v$_{q}$ = ∇$_{q}$ ω$_{q}$ = 2 W π . (29) As shown in Figs. 3c, 4c and 5c, the angle between the two trajectories of phonon wave packets departing from point (0 , 0) but with opposite velocities in Figs. 3c and 5c ( W = 0 . 8) is about 8 times of that in Fig. 4c ( W = 0 . 1). For a larger value of transfer integral, e.g., J = 1 . 0 in Fig. 5, the left-moving and right-moving wave packets of the exciton depart the site of creation and makes a quick rendezvous at the opposite site of the ring ( n = 16), where the recombined exciton density remains sufficiently high to trigger another pair of localized phonon wave packets as is clearly demonstrated in Figs. 5b and 5c. The exciton will travel at a much reduced speed after the rendezvous at n = 16, and further stimulations of phonon wave packets are all but impossible for J = 1 . 0. C. Vibrational Amplitude of Exciton and Phonon using ˜ D Ansatz In Figs. 6, 7 and 8, time evolution of the variational pa- rameters | ψ$_{n}$ ( t ) | $^{2}$, | λ$_{n}$ ( t ) | and | λ$_{n}$ ( t ) | is displayed for the ˜ D Ansatz. When J and W are both small, the phonons are mostly localized in the vicinity of the site where the exciton is initially created. The first set of phonon dis- placements, | λ$_{n}$ ( t ) | , will extend slightly to the neighbor- ing sites as the time increases because of a nonzero J and a finite phonon dispersion (Fig. 7a). But as shown in Fig. 8a, there is almost no spreading of β$_{n}$ ( t ), sug- gesting that this portion of phonon displacements closely follows the exciton and appears almost entirely on the site of exciton creation. Fig. 8a also reveals oscillatory behavior of | β$_{n}$ ( t ) | : when t equals (2 n + 1) π/ω$_{0}$ , | β$_{n}$ ( t ) | reaches its maximum value; and when t equals 2 nπ/ω$_{0}$ , As J is incre ased, the exciton will have the ability to move away from the site of creation. As shown in Figs. 6b and 6d, the lar g er J is, the faster the exciton transfers. Although J ha s no direct influence over the phonon dis- placements, a la rge J induces more excitonic coherence between adjace nt sites, and in turn, causes more phonon deformations on those sites (cf. Fig. 7). The propagation the amplitude is at its minimum. Because the transfer integral J is ne arly zero, the exciton stays very close to the site of creat ion. From Fig. 6a, it is obvious that the exciton stays lo calized on one site with hardly any prop- agation for a s mall value of transfer integral J = 0 . 1. In this case, it see ms that oscillations of phonon amplitudes also influence t he movement of the exciton. The propa- gation of the ex citon from the n = 0 site to its nearest neighbors occur s only at t = 2 nπ/ω$_{0}$ when the phonon amplitude reach es a minimum value. Otherwise, because of the coupling with the phonons, the exciton will be trapped and un able to transfer to other sites. FIG. 5: Time e volution of variation parameters of the D 2 Ansatz for J = 1 . 0 , W = 0 . 8 and S = 0 . 5. (a) A bird’s- eye view of the e xciton probability | ψ$_{n}$ ( t ) | 2 in real space; (b) a contour plot of | ψ$_{n}$ ( t ) | $^{2}$; (c) a contour plot of the phonon displacement | λ$_{n}$ ( t ) | in real space. −15 −10 −5 0 5 10 15 0 2 4 6 0 0.2 0.4 0.6 0.8 1 t/ (2 π/ω 0 ) n  (a) t/ (2 π/ω 0 )  (b) 0 1 2 3 4 5 6 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 n t/ (2 π/ω 0 )  (c) −15 −10 −5 0 5 10 15 0 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35