  The non-abelian Hodge correspondence identifies complex variations of Hodge
structures with certain Higgs bundles. In this work we analyze this
relationship, and some of its ramifications, when the variations of Hodge
structures are determined by a (complete) one-dimensional family of compact
Calabi-Yau manifolds. This setup enables us to apply techniques from mirror
symmetry. For example, the corresponding Higgs bundles extend to parabolic
Higgs bundles to the compactification of the base of the families. We determine
the parabolic degrees of the underlying parabolic bundles in terms of the
exponents of the Picard-Fuchs equations obtained from the variations of Hodge
structure. Moreover, we prove in this setup that the flat non-abelian Hodge or
$tt^*$-connection is gauge equivalent to an oper which is determined by the
corresponding Picard-Fuchs equations. This gauge equivalence puts forward a new
derivation of non-linear differential relations between special functions on
the moduli space which generalize Ramanujan's relations for the differential
ring of quasi-modular forms.
