  Pair operators for boson and fermion atoms generate SU(1,1) and SU(2) Lie
algebras, respectively. Consequently, the pairing of boson and fermion atoms
into diatomic molecules via Feshbach resonances, produces SU(1,1) and SU(2)
coherent states, making bosonic pairing the matter-wave equivalent of
parametric coupling and fermion pairing equivalent to the Dicke model of
quantum optics. We discuss the properties of atomic states generated in the
dissociation of molecular Bose-Einstein condensates into boson or fermion
constituent atoms. The SU(2) coherent states produced in dissociation into
fermions give Poissonian atom-number distributions, whereas the SU(1,1) states
generated in dissociation into bosons result in super-poissonian distributions,
in analogy to two-photon squeezed states. In contrast, starting from an atomic
gas produces coherent number distributions for bosons and super-poissonian
distributions for fermions.
