Quantum theory of atomic four-wave mixing in Bose-Einstein condensates

We present an exact quantum-mechanical analysis of collinear four-wave mixing in a multicomponent Bose-Einstein condensate consisting of sodium atoms in the $F=1\mathrm{}$ ground state. Technically, this is achieved by taking advantage of the conservation laws of the system to cast its Hamiltonian in terms of angular momentum operators. We discuss explicitly the build-up of matter-wave side modes from quantum fluctuations, as well as the correlations between these modes. We show the appearance of a strong quantum entanglement between hyperfine states. We also demonstrate that for finite atomic numbers, the system exhibits periodic collapses and revivals in the exchange of atoms between different spin states.