Analyticity of smooth eigenfunctions and spectral analysis of the Gauss map

We provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form Uf(x) = integral a(x, y) f(b( x, y)) mu(dy) acting on functions f: [u, v] --> C ( evolution operators of one-dimensional dynamical systems and Markov processes have this form). We estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the Frobenius-Perron operator of the continuous fraction Gauss map. We prove that any infinitely differentiable eigenfunction f of this Frobenius-Perron operator, corresponding to a non-zero eigenvalue admits a (unique) analytic extension to the set C\(-infinity, 1]. Analyzing the spectrum of the Frobenius Perron operator in spaces of smooth functions, we extend significantly the domain of validity of the Mayer and Ropstorff asymptotic formula for the decay of correlations of the Gauss map.

Vortex entanglement in Bose-Einstein condensates coupled to Laguerre-Gauss beams

We study the establishment of vortex entanglement in remote Bose-Einstein condensates (BECs). We consider a two-mode photonic resource entangled in its orbital angular momentum (OAM) degree of freedom and, by exploiting the process of light-to-BEC OAM transfer, demonstrate that such entanglement can be efficiently passed to the matterlike systems. Our proposal thus represents a building block for novel dissipation-free and long-memory communication channels based on OAM. We discuss issues of practical realizability, stressing the feasibility of our scheme, and present an operative technique for the indirect inference of the set vortex entanglement.