  We show that the cold atom systems of simultaneously trapped Bose-Einstein
condensates (BEC's) and quantum degenerate fermionic atoms provide promising
laboratories for the study of macroscopic quantum tunneling. Our theoretical
studies reveal that the spatial extent of a small trapped BEC immersed in a
Fermi sea can tunnel and coherently oscillate between the values of the
separated and mixed configurations (the phases of the phase separation
transition of BEC-fermion systems). We evaluate the period, amplitude and
dissipation rate for $^{23}$Na and $^{40}$K-atoms and we discuss the
experimental prospects for observing this phenomenon.
