  Optical control of structure-driven magnetic order offers a platform for
magneto-optical terahertz devices. We control the magnetic phases of $d^1$ Mott
insulating titanates using nonlinear phononics to transiently perturb the
atomic structure based on density functional theory (DFT) simulations and
solutions to a lattice Hamiltonian including nonlinear multi-mode interactions.
We show that magnetism is tuned by indirect excitation of a Raman-active phonon
mode, which affects the amplitude of the TiO$_6$ octahedral rotations that
couple to static Ti--O Jahn-Teller distortions, through infrared-active phonon
modes of LaTiO$_3$ and YTiO$_3$. The mode excitation reduces the rotational
angle, driving a magnetic phase transition from ferromagnetic (FM) to $A$-type
antiferromagnetic (AFM), and finally a $G$-type AFM state. This novel $A$-AFM
state arises from a change in the exchange interactions and is absent in the
bulk equilibrium phase diagram, but it emerges as a dynamically accessible
optically induced state under multi-mode excitations. Our work shows nonlinear
phononic coupling is able to stabilize phases inaccessible to static chemical
pressure or epitaxial strain.
