Inducing Ferroquadrupolar Order with Applied Magnetic Field in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>TbPO</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> </m

In this work, we demonstrate a remarkable field-induced ferroquadrupolar phase emanating from the magnetoelastic coupling. Using synchrotron x-ray and neutron diffraction, heat capacity measurements, as well as mean-field modeling, we show that above the critical magnetic field, ferroquadrupolar <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msubsup> <a:mi>O</a:mi> <a:mn>2</a:mn> <a:mrow> <a:mo>−</a:mo> <a:mn>2</a:mn> </a:mrow> </a:msubsup> </a:math> order is stabilized as the main order parameter, in competition with the field-induced magnetic polarization. This is revealed by a severe lattice distortion and the suppression of antiferromagnetic ordering of the dipole moments. Even in zero magnetic field, the previously reported antiferromagnetic order is a mixed quadrupolar-dipolar phase, further highlighting the crucial role of spin-lattice interactions. This complete understanding of the magnetoelastic phase diagram establishes <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:msub> <c:mrow> <c:mi>TbPO</c:mi> </c:mrow> <c:mrow> <c:mn>4</c:mn> </c:mrow> </c:msub> </c:mrow> </c:math> as a model system for studying nematic-antiferromagnetic transitions and provides key Hamiltonian parameters for its description.