The non-Abelian gauge fields play a key role in achieving novel quantum phenomena in condensed-matter and high-energy physics. Recently, the synthetic non-Abelian gauge fields have been created in the neutral degenerate Fermi gases, and moreover, gen
erate many exotic effects. All the previous predictions can be well understood by the microscopic Bardeen-Cooper-Schrieffer theory. In this work, we establish an SU(2) Ginzburg-Landau theory for degenerate Fermi gases with the synthetic non-Abelian gauge fields. We firstly address a fundamental problem how the non-Abelian gauge fields, imposing originally on the Fermi atoms, affect the pairing field with no extra electric charge by a local gauge-field theory,and then obtain the first and second SU(2) Ginzburg-Landau equations. Based on these obtained SU(2) Ginzburg-Landau equations, we find that the superfluid critical temperature of the intra- (inter-) band pairing increases (decreases) linearly, when increasing the strength of the synthetic non-Abelian gauge fields. More importantly, we predict a novel SU(2) non-Abelian Josephson effect, which can be used to design a new atomic superconducting quantum interference device.
