Matter-wave squeezing and the generation of SU(1,1) and SU(2) coherent states via Feshbach resonances

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.