Polarized neutron diffraction experiments have been performed on multiferroic materials $R$Mn$_{2}$O$_{5}$ ($R=$Ho, Er) under electric fields in the ferroelectric commensurate (CM) and the low-temperature incommensurate (LT-ICM) phases, where the former has the highest electric polarization and the latter has reduced polarization. It is found that, after cooling in electric fields down to the CM phase, the magnetic chirality is proportional to the electric polarization. Also we confirmed that the magnetic chirality can be switched by the polarity of the electric polarization in both the CM and LT-ICM phases. These facts suggest an intimate coupling between the magnetic chirality and the electric polarization. However, upon the transition from the CM to LT-ICM phase, the reduction of the electric polarization is not accompanied by any reduction of the magnetic chirality, implying that the CM and LT-ICM phases contain different mechanisms of the magnetoelectric coupling.
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