Field-induced electric polarization and elastic softening caused by parity-mixed $d$-$p$ hybridized states with electric multipoles in Ba$_2$CuGe$_2$O$_7$

We performed high-magnetic-field magnetization, polarization, and ultrasonic measurements in Ba$_2$CuGe$_2$O$_7$ to investigate field-induced multiferroic properties arising from a cross-correlation between electric dipoles and electric quadrupoles in addition to cross-correlation between magnetic dipoles and electric dipoles. Magnetization $M$ shows saturation behavior above 20 T for several magnetic field directions, however, electric polarization $P_c$ exhibits an increase, and elastic constants show a softening above 20 T. Based on quantum states with a crystalline electric field for the $D_{2d}$ point group and $d$-$p$ hybridization between Cu-$3d$ and O-$2p$ electrons, we confirmed that the matrix of an electric dipole $P_z$ was proportional to that of an electric quadrupole $O_{xy}$. Furthermore, considering the spin-orbit coupling of $3d$ electrons and the Zeeman effect, we showed that $P_z$ and $O_{xy}$ simultaneously exhibited field-induced responses. These findings indicate that the orbital degrees of freedom, in addition to the spin degrees of freedom, contribute to the high-field multiferroicity in Ba$_2$CuGe$_2$O$_7$.