Motional Broadening in Ensembles with Heavy-Tail Detuning Distribution
In previous chapters we have shown that fluctuations in the resonance frequency of the two-level systems (TLS) cause narrowing of the spectrum. In other words, we have shown that fluctuations change the time-evolution of the phase difference between the two levels of the TLS from linear (ballistic) expansion to diffusion. In this chapter we address the question under what conditions the fluctuations can have the reverse effect to motional narrowing and lead to broadening of the spectral lines. An example for this effect was pointed out in References. We analyze the problem in a spectroscopic framework, and show that when the ensemble frequency distribution has heavy tails with a diverging mean, motional broadening emerges. In terms of quantum information, this manifests itself as a shortening of the coherence time as the the fluctuation rate increases. We derive a general equation for the linewidth of the spectrum, and demonstrate its validity through numerical simulations. Since in practice heavy tails of the frequency distribution can be sustained only up to some point, we study scenarios with cutoffs and show that motional broadening persists up to some fluctuation rate. Motional broadening is relevant to many fields in which heavy-tail distributions are encountered, including turbulence, diffusion and laser-cooling.
KeywordsHeavy Tail Stable Distribution Coherence Time Fluctuation Rate Cauchy Distribution
- 5.Feller W (1968) An introduction to probability theory and its applications, vol II, 3rd edn. Wiley, New YorkGoogle Scholar
- 6.Ibragimov IA, Linnik YV (1971) Independent and stationary sequences of random variables. Wolters-Noordhoff, GroningenGoogle Scholar
- 9.Castin Y, Dalibard J, Cohen-Tannoudji C (1991) The limits of Sisyphus cooling. In: Moi L, Gozzini S, Gabbanini C, Arimondo E, Strumia F (eds.) Light induced kinetic effects on atoms, ions, and molecules. ETS Editrice, PisaGoogle Scholar