Mesoscopic superfluid to superconductor transition

Spectrum tomography for the energy ($E$) of a ring-shaped Bose-Hubbard circuit is illustrated. There is an inter-particle interaction $U$ that controls superfluidity (SF) and the transition to the Mott Insulator (MI) regime. The circuit is coupled to an electromagnetic cavity mode of frequency $\omega_0$, and the coupling is characterized by a generalized fine-structure-constant $\alpha$ that controls the emergence of superconductivity (SC). The ${(U,\alpha,\omega_0,E)}$ diagram features SF and SC regions, a vast region of fragmented possibly chaotic states, and an MI regime for large $U$. The mesoscopic version of the Meissner effect and the Anderson-Higgs mechanism are discussed.