Distribution of variances of single molecules in a disordered lattice

Single molecule spectroscopy in disordered lattice is investigated from a t heoretical point of view. We consider energy fluctuations of the molecule d ue to dipolar interactions with two level system (TLS) defects distributed randomly on a three-dimensional cubic lattice. Each independent TLS is rand omly nipping so the energy of the molecule is time dependent. We investigat e the probability distribution g(W) of the variance of the energy fluctuati ons. The exact solution, found for finite systems, exhibits peaks, peaks wi thin peaks, etc., corresponding to interaction with nearest neighbors, next -nearest neighbors, etc. For an infinite crystal at high defect density, th e distribution of W is shown to depend strongly on the interaction with nea rest neighbors and hence on lattice symmetry. At low defect density, g(W) e xhibits several peaks separated by large gaps in which g(W) similar to 0. W e explain these peaks in terms of contributions from single defects located on discrete distances from the molecule. For the continuum version of our model, g(W) is a Levy stable nonsymmetrical probability density function, d ecaying according to g(W) similar to W-3/2. We discuss the relation between the continuum and lattice models.