  Recent experimental studies have demonstrated the possibility of utilizing
strong terahertz pulses to manipulate individual ferroic orders on pico- and
femtosecond timescales. Here, we extend these findings and showcase the
simultaneous manipulation of multiple ferroic orders in BiFeO$_3$, a material
that is both ferroelectric and antiferromagnetic at room temperature. We find a
concurrent enhancement of ferroelectric and antiferromagnetic second-harmonic
generation (SHG) following the resonant excitation of a high-frequency
fully-symmetric phonon mode. Based on first-principles calculations and
phenomenological modeling, we ascribe this observation to the inherent coupling
of the two ferroic orders to the nonequilibrium distortions induced in the
crystal lattice by nonlinearly driven phonons. Our finding highlights the
potential of nonlinear phononics as a technique for manipulating multiple
ferroic order parameters at once. In addition, this approach provides a
promising avenue to studying the dynamical magnetization and polarization
behavior, as well as their intrinsic coupling, on ultrashort timescales.
