Reduced fluctuations for bosons in a double well

We review two recent results on the ground state properties of bosonic systems trapped by a double well external potential. In the limit of a large number of particles and large separation between the wells, we prove that fluctuations in the number of particles occupying each single-well low-energy mode occur at a reduced scale with respect to N, the latter being the typical prediction of the central limit theorem. This signals the breakdown of the independent and uncorrelated particle picture of standard Bose-Einstein condensation and the emergence of an interaction-driven correlated phase in the ground state.