  High-harmonic generation (HHG), a typical nonlinear optical effect, has been
actively studied in electron systems such as semiconductors and
superconductors. As a natural extension, we theoretically study HHG from
electric polarization, spin current and magnetization in magnetic insulators
under terahertz (THz) or gigahertz (GHz) electromagnetic waves. We use simple
one-dimensional spin chain models with or without multiferroic coupling between
spins and the electric polarization, and study the dynamics of the spin chain
coupled to an external ac electric or magnetic field. We map spin chains to
two-band fermions and invoke an analogy of semiconductors and superconductors.
With a quantum master equation and Lindblad approximation, we compute the time
evolution of the electric polarization, spin current, and magnetization,
showing that they exhibit clear harmonic peaks. We also show that the
even-order HHG by magnetization dynamics can be controlled by static magnetic
fields in a wide class of magnetic insulators. We propose experimental setups
to observe these HHG, and estimate the required strength of the ac electric
field $E_0$ for detection as $E_0\sim100$kV/cm--1MV/cm, which corresponds to
the magnetic field $B_0\sim0.1$T--1T. The estimated strength would be relevant
also for experimental realizations of other theoretically-proposed nonlinear
optical effects in magnetic insulators such as Floquet engineering of magnets.
