Origin of the magnetic-field controlled polarization reversal in multiferroic TbMn$_{2}$O$_{5}$

The interplay of multi-dimensional complex magnetic order parameters leads to interesting effects like magnetically induced ferroelectricity. A particular interesting example is TbMn$_{2}$O$_{5}$ because of the associated magnetic-field controllable electric polarization. By optical second harmonic generation we show that the gigantic magnetoelectric effect originates in three independent ferroelectric contributions. Two of these are manganese-generated. The third contribution is related to the magnetism of the Tb$^{3+}$ sublattice and has not been identified so far. It mediates the remarkable magnetic-field induced polarization reversal. This model is verified by experiments on the isostructural YMn$_{2}$O$_{5}$ where Y$^{3+}$ ions are nonmagnetic and only two polarization contributions are present and no magnetoelectric coupling is observed. These results underline the importance of the $3d-4f$-interaction for the intricate magnetoelectric coupling in the class of isostructural RMn$_{2}$O$_{5}$ compounds.