Experimental evidence for Lévy statistics in single-molecule spectroscopy in a low-temperature glass-manifestation of long-range interactions

We demonstrate that the statistical behavior of the random line shapes of single tetra-tert-butylterrylene chromophores embedded in an amorphous polyisobutylene matrix at T=2 K is described by Lévy statistics. Recently, Barkai et al., suggested to characterize random line shapes of single molecules in glasses by their cumulants κ₁,κ ₂,κ₃, etc. Using Geva-Skinner model for single-molecule spectroscopy in low-temperature glasses, which is based on the standard tunneling model, the theory predicts that probability densities P(κ₁), P(κ₂), etc., are Lévy stable laws provided that the glass dynamics is described by a slow modulation limit. Analyzing our experimental data we show that the distributions of the first two cumulants are indeed compatible with Lévy statistics; thus, the generalized central limit theorem is applicable to this system. The emergence of Lévy stable laws in this system is due to long-range interactions between two-level systems and the single molecule. The widths of the distribution functions P(κ₁) and P(κ₂) are non-universal in the sense that they depend on the coupling of the molecule to the host glass. We investigate a universal amplitude ratio (i.e., ratio of widths) which shows that our results are in agreement with the assumptions of the standard tunneling model of low-temperature glasses. We briefly discuss other long-range interacting systems and models, for which Lévy statistics plays an important role.