Motional Broadening in Ensembles With Heavy-Tail Frequency Distribution

Motional narrowing is a well-known phenomenon in which the spectrum is narrowed because the frequency of each particle in the ensemble is fluctuating. Motional narrowing was observed in many physical systems, ranging from liquid NMR [1] to cold atomic ensembles [2]. Here we show that fluctuations can actually have the reverse effect and lead to broadening of the spectrum. We prove that the condition for this to happen is that the ensemble frequency distribution will have heavy tails with a diverging mean. We also show that for both motional narrowing and broadening the asymptotic decay of the coherence in exponential, and derive an expression for the decay rate. Finally, we study a scenario with a cutoff in the frequency distribution heavy-tail and find that motional broadening still persists up to some fluctuation rate. \\[4pt] [1] Bloembergen, N., Purcell, E. M., and Pound, R. V. Phys. Rev. 73, 679 (1948). \newline [2] Sagi, Y., Almog, I., and Davidson, N. Phys. Rev. Lett. 105, 093001 (2010).