We have found that superconductivity and superfluidity have a connection to quantum geometry. Namely, the superfluid weight in a multiband system has a previously unnoticed component which we call the geometric contribution. It is proportional to the minimal quantum metric of the band. Using this theory, we have shown that superconductivity is possible also in a flat band where individual electrons would not move, possibly relevant to twisted bilayer graphene. Also, the quantum transport in flat band shows unique behavior. We have also explored the effect of quantum geometry on Bose-Einstein condensation (BEC). Arrays of plasmonic nanoparticles, so-called plasmonic lattices, when combined with an emitter material (gain medium), provide a versatile platform for studies on light-matter interaction in the nanoscale, including collective coherent phenomena as well as topological photonics. We have experimentally realized a BEC of hybrids of surface plasmons and light in a nanoparticle array. In the lasing regime, we have observed bound state in continuum modes with different topological charges, even charges as high as 19 in quasicrystal geometries. W also studied the quantum geometric tensor in these systems. We experimentally observed non-zero quantum metric and Berry curvature along the diagonals of the Brillouin zone of a square lattice of gold nanoparticles. We show that the Berry curvature originates solely from non-Hermitian effects.
