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 m_{F} = -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. Similar behavior (spin waves) have been observed in thermal (not Bose-condensed) clouds, which is remarkable inasmuch as the spatial degrees of freedom of the atoms are all different in the thermal cloud, while they are all the same in a BEC. We are now studying how partially-Bose-condensed systems behave in this context as well. We have also carried on long-standing studies of photoassociation spectroscopy with laser-cooled sodium. Photoassociation spectroscopy is a sensitive probe of the cold atoms and also provides a handle with which to manipulate the atom-atom interactions.

## Summary

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 measurements of the spin 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. We are currently studying similar spin dynamics in clouds of thermal atoms and mixtures of thermal and Bose-condensed atoms as well.

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Collaborators

Eite Tiesinga (NIST)

Ryan Wilson (U.S. Naval Academy)