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,κ2,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.