Abstract
We study the dynamical interplay between ferroelectricity and magnetism in a multiferroic with a helical magnetic order. We show that the dynamical exchange-striction induces a biquadratic interaction between the spins and transverse phonons resulting in quantum fluctuations of the spontaneous ferroelectric polarization P in the ferroelectric phase. The hybridization between the spin wave and the fluctuation of the electric polarization leads to low-lying transverse phonon modes. Those are perpendicular to P and to the helical spins at small wave vector but then turn parallel to P at a wave vector close to the magnetic modulation vector. For helical magnetic structure, the spin chirality which determines the direction of P, also possesses a long-range order. Due to the dynamical Dzyaloshiskii-Moriya interaction, the spin chirality is strongly coupled to the spin fluctuation which implies an on-site inversion of the spin chirality in the ordered spin-(1/2) system and results in a finite scattering intensity of polarized neutrons from a cycloidal helimagnet.