Distribution of Variances of Single Molecules in a Disordered Lattice

Single molecule spectroscopy in disordered lattice is investigated from a theoretical point of view. We consider energy fluctuations of the molecule due to dipolar interactions with two level system (TLS) defects distributed randomly on a three-dimensional cubic lattice. Each independent TLS is randomly flipping so the energy of the molecule is time dependent. We investigate the probability distribution g(W) of the variance of the energy fluctuations. The exact solution, found for finite systems, exhibits peaks, peaks within peaks, etc., corresponding to interaction with nearest neighbors, next-nearest neighbors, etc. For an infinite crystal at high defect density, the distribution of W is shown to depend strongly on the interaction with nearest neighbors and hence on lattice symmetry. At low defect density, g(W) exhibits several peaks separated by large gaps in which g(W) ∼ 0. We 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, decaying according to g(W) ∼ W^-3/2. We discuss the relation between the continuum and lattice models.