Theory of Rayleigh scattering using finite-dimension random-matrix theory

We describe in terms of random matrices of finite dimension N the time-dependent resonant Rayleigh scattering (RRS), which is a powerful tool to study the quantum mechanics of localization in disordered systems. Three contributions to the lifetime corrected RRS signal can be distinguished by their temporal (and angular) dependence: During an initial time span given by the inverse inhomogeneous linewidth, the specularly reflected beam decreases in intensity. Simultaneously, a broad background and enhanced emission in the backward direction start quadratically. On a longer time scale, enhanced emission into the forward direction comes up, reaching finally the same value as the backward emission. In a situation where an angular resolution is not possible, the model is extended towards an ensemble of uncorrelated random matrix subsystems. Then, the emission into nonspecular directions reaches soon a universal value of 2/3 of the final signal. The subsequent slow increase can be traced back to the peculiar longer time scale of the forward scattering part. This scale is set by the inverse energy distance of spatially overlapping eigenstates, and consequently called level repulsion time $t_{\mathrm{LR}}.$ The time dependence itself is given by the random-matrix theory form factor, i.e., by the Fourier-transformed level correlation function. The exact results derived for this description compare favorably with large-scale numerical simulations for a realistic exciton model with specific material parameters.