  We develop a microscopic model to describe the Josephson dynamics between two
superfluid reservoirs of ultracold fermionic atoms which accounts for the
dependence of the critical current on both the barrier height and the
interaction strength along the crossover from BCS to BEC. Building on a
previous study [F. Meier & W. Zwerger, Phys. Rev. A, 64 033610 (2001)] of
weakly-interacting bosons, we derive analytic results for the Josephson
critical current at zero temperature for homogeneous and trapped systems at
arbitrary coupling. The critical current exhibits a maximum near the unitarity
limit which arises from the competition between the increasing condensate
fraction and a decrease of the chemical potential along the evolution from the
BCS to the BEC limit. Our results agree quantitatively with numerical
simulations and recent experimental data.
