  Non-Abelian gauge field (NAGF) plays a central role in understanding the
geometrical and topological phenomena in physics. Here we experimentally induce
a NAGF in the degenerate eigen subspace of a double-$\Lambda$ four-level atomic
system. The non-Abelian nature of the gauge field is detected through the
measurement of the non-commutativity of two successive evolution loops.
  Then we theoretically propose and experimentally demonstrate a novel scheme
to measure the NAGF through multi-loop evolution and robust holonomic quantum
gates. The demonstrated scheme offers the advantage of detecting the NAGF with
amplification through multi-loop evolution. Our results pave the way for an
experimentally-feasible approach to achieving high-resolution and
high-precision measurements of the gauge fields.
