  We consider the isoperimetric problem defined on the whole $\mathbb{R}^n$ by
the Allen--Cahn energy functional. For non-degenerate double well potentials,
we prove sharp quantitative stability inequalities of quadratic type which are
uniform in the length scale of the phase transitions. We also derive a rigidity
theorem for critical points analogous to the classical Alexandrov's theorem for
constant mean curvature boundaries.
