In the last years the physics of ultra-cold spinor gases (atoms
with more than two internal degrees of freedom, i.e. spin or pseudo-spin
higher than 1/2) has attracted a large interest. In this
talk I will first review some of the basic features of ultra-cold spinor
gases, and in particular how the interplay between interatomic interactions,
Zeeman energy and trapping energy induces a rich physics in these gases,
both in what concerns their ground-state phases and the spinor dynamics.
I will then present some of our recent results on ultra-cold spinor gases.
In particular, I will first discuss parametric
amplification of spin-vacuum fluctuations in spinor Bose-Einstein condensates.
I will then comment on the properties of strongly-correlated fermions
in one-dimensional optical lattices. I will specifically show how the
quadratic Zeeman effect may induce various types of quantum phase transitions
between various magnetic phases of a Mott-insulator of spin-3/2 fermions.