Nonlinear spin current of photoexcited magnons in collinear antiferromagnets

We study the nonlinear magnon spin current induced by an ac electric field under light irradiation in collinear antiferromagnets with broken inversion symmetry. For linearly polarized light, we find that a dc spin current appears through ``the magnon spin shift current"mechanism, which is driven by a spin polarization generation in the two magnon creation process and has a close relationship to the geometry of magnon bands through Berry connection. For circularly polarized light, a dc spin current appears through ``the spin injection current"mechanism, which is proportional to the relaxation time of magnons and can be large when the magnon lifetime is long. We demonstrate generation of the magnon spin shift and injection currents, based on a few toy models and a realistic model for a multiferroic material M$_2$Mo$_3$O$_8$.