Intermediate scattering function of colloids in a periodic laser field

We investigate the dynamics of individual colloidal particles in a one-dimensional periodic potential using the intermediate scattering function (ISF) as a key observable. We elaborate a theoretical framework and derive formally exact analytical expressions for the ISF. We introduce and analyze a generalized ISF with two wave vectors to capture correlations in a periodic potential beyond the standard ISF. Relying on Bloch's theorem for periodic systems and, by solving the Smoluchowski equation for an overdamped Brownian particle in a cosine potential, we evaluate the ISF by numerically solving for the eigenfunctions and eigenvalues of the expression. We apply time-dependent perturbation theory to expand the ISF and extract low-order moments, including the mean-square displacement, the time-dependent diffusivity, and the non-Gaussian parameter. Our analytical results are validated through Brownian-dynamics simulations and experiments on 2D colloidal systems exposed to a light-induced periodic potential generated by two interacting laser beams.