  To describe the tunneling dynamics of a stack of two-dimensional fermionic
superfluids in an optical potential, we derive an effective action functional
from a path integral treatment. This effective action leads, in the saddle
point approximation, to equations of motion for the density and the phase of
the superfluid Fermi gas in each layer. In the strong coupling limit (where
bosonic molecules are formed) these equations reduce to a discrete nonlinear
Schrodinger equation, where the molecular tunneling amplitude is reduced for
large binding energies. In the weak coupling (BCS) regime, we study the
evolution of the stacked superfluids and derive an approximate analytical
expression for the Josephson oscillation frequency in an external harmonic
potential. Both in the weak and intermediate coupling regimes the detection of
the Josephson oscillations described by our path integral treatment constitutes
experimental evidence for the fermionic superfluid regime.
