  The internal Josephson oscillations between an atomic Bose-Einstein
condensate (BEC) and a molecular one are studied for atoms in a square optical
lattice subjected to a staggered gauge field. The system is described by a
Bose-Hubbard model with complex and anisotropic hopping parameters that are
different for each species, i.e., atoms and molecules. When the flux per
plaquette for each species is small, the system oscillates between two
conventional zero-momentum condensates. However, there is a regime of
parameters in which Josephson oscillations between a vortex-carrying atomic
condensate (finite momentum BEC) and a conventional zero-momentum molecular
condensate may be realized. The experimental observation of the oscillations
between these qualitatively distinct BEC's is possible with state-of-the-art
Ramsey interference techniques.
