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## A. Reduced Exciton Density Matrix and Coherence Size 

The reduced single-exciton density matrix ρ$_{mn}$ ( t ) can be obtained after solving the coupled equations of varia- tional parameters 

where ρ ( t ) = | Φ( t ) 〉〈 Φ( t ) | is the full density matrix at zero temperature, and | Φ( t ) 〉 is the total polaron wave function at time t after the photo excitation takes place. Substituting the detailed form of the D$_{2}$ Ansatz into the polaron wave function, one obtains 

For the D$_{2}$ Ansatz, the reduced exciton density matrix involves only the exciton amplitude, and does not contain information of the phonon manifold. In contrast, the more sophisticated trial state, the ˜ D Ansatz, includes the Debye-Waller factor S$_{n,m}$ in the expression of the reduced exciton density matrix: 

where S$_{n,m}$ can be written as 

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## III. RESULTS AND DISCUSSIONS 

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Using the Merrifield Ansatz $^{8}$, when J = 0, solutions to coupled equations of variational parameters can be ob- tained analytically for zero phonon bandwidth ( W = 0). It is found that L$_{ρ}$ = 1 /N for all t = 2 πn/ω$_{0}$ ( n = 1 , 2 , 3 , ... ). Periodic changes of the polaron structure as a function of time have also been revealed when J and W are both small. At ω$_{0}$t = 7 π, 8 π, 9 π and 10 π , the re- duced exciton density matrix ρ$_{mn}$ calculated from the ˜ D Ansatz for the case of J = 0 . 1 , W = 0 . 1 and S = 0 . 5 is displayed in Fig. 1. At t = 0, ρ$_{mn}$ is confined to the (0 , 0) point, i.e., the exciton is localized at one site of the one-dimensional system. Then, as the excited state re- laxes, the non-zero elements of ρ$_{mn}$ spread out gradually. Oscillatory behavior is visible during the evolution of the density matrix as both the J and W are small, and as a result, dephasing due to phonon dispersion and exciton transfer is at a minimal level. When t equals to 2 nπ/ω$_{0}$ , the matrix is rather delocalized and have a symmetric distribution, and at t = (2 n + 1) π/ω$_{0}$ , matrix elements aggregate along the diagonal line. 

When J is increased, the oscillatory behavior of the density matrix gradually fades as the exciton density de- localization happens more swiftly. From the reduced ex- citon density matrix, one can obtain an exciton coherence size L$_{ρ}$ 32–34 characterizing the size of a domain within which the chromophores emit coherently. All informa- tion relevant to the excitonic superradiance is contained within the reduced exciton density matrix ρ$_{mn}$ ( t ), the time dependence of which provides a clear view of the relaxation process. Previously, for translationally invari- ant states, a definition of the characteristic coherence size L$_{ρ}$ in terms of the density matrix was introduced by ap- plying the inverse participation ratio concept commonly used in the theory of quantum localization: 

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This definition of the coherence size is borrowed to quan- 

