Interactions between cold atoms can provide us with the means to manipulate matter on the smallest scales. Collisions between laser-cooled atoms, in particular amongst those in Bose-Einstein condensates, can allow the coherent manipulation of clouds of atoms. We study spinor condensates, where the atoms can be in a superposition of internal quantum states and investigate the internal-state dynamics of the cloud, as well as studying photoassociation reactions where colliding cold atoms absorb a photon to form an excited diatomic molecule. The goals of such studies are to, for example, form spin-squeezed states with noise on spin measurements reduced below the standard quantum limit. In the long term clouds of atoms in such a state can provide for improved measurements of, for instance, magnetic fields.
A Bose-Einstein condensate (BEC) collapses the wavefunctions of many particles in to a single quantum state. In a spinor BEC the atoms can be in a superposition of internal quantum states. Thus, a BEC of spin-1 particles, like the F = 1 ground state of Na atoms, can be thought of as being a single condensate with the atoms in a superposition of the three spin projections mF = -1, 0, and +1, or equivalently, the superposition of three coupled BECs with the same spatial wavefunction, one in each of these spin states. This many-particle system can then be described by a single wavefunction that is a product of the spatial distribution and an internal wavefunction that is a mathematical spinor, or vector wavefunction describing the populations and phases of the coupled BEC components. The dynamics and steady states of the components of the spinor wavefunction can then be described with a relatively simple set of equations that in this case involve only the collisional interactions amongst the different spin states and the quadratic Zeeman or magnetic field interaction. The system can be described using only a single population and a single phase, exchanging populations only through the collisions that convert two m = 0 atoms into a m = +1 and a m = -1 pair. We have studied the dynamics of F = 1 Na spinor BECs that are created in non-equilibrium states and verified the "single spatial mode approximation" theory that has been developed to describe the population oscillations at short times. At long times the system finds its steady state and can be driven through quantum phase transitions as indicated in the figure. The system can be either a 2-component BEC with all of the m = 0 atoms converted to +1/-1 pairs, or a 3-component BEC with all of the spin projections populated. The transition from 2- to 3-component condensate is driven by changing either the magnetic field, B, or the net magnetization (excess of m = +1 over m = -1 population) of the system. The transition is a quantum phase transition in that it is not driven by thermal fluctuations but rather quantum fluctuations. The dissipative dynamics that occur between the initial non-equilibrium states and the final steady states of the system are now an active topic of further investigation. We have also carried on long-standing studies of photoassociation spectroscopy with laser-cooled sodium. If one assumes some knowledge of the molecular spectra one can learn much about the collision dynamics. If, on the other hand, one understands the collisions to some degree, one can learn the details of the molecular spectroscopy. The resolution of such spectra are often much higher than are available from molecular beam spectroscopy, as the energy spread of the colliding atoms is extremely small - often smaller than the natural width of the lines being studied. Photoassociation spectroscopy is a sensitive probe of the cold atoms and also provides a handle with which to manipulate the atom-atom interactions.