  We propose to subject two Bose-Einstein condensates to a periodic potential,
so that one condensate undergoes the Mott insulator transition to a state with
precisely one atom per lattice site. We show that photoassociation of
heteronuclear molecules within each lattice site is described by the quantum
optical Jaynes-Cummings Hamiltonian. In analogy with studies of this
Hamiltonian with cavity fields and trapped ions, we are thus able to engineer
quantum optical states of atomic matter wave fields and we are able to
reconstruct these states by quantum state tomography.
