  The suppression of ferroquadrupolar order in TmVO$_4$ in a magnetic field is
well-described by the transverse field Ising model, enabling detailed studies
of critical dynamics near the quantum phase transition. We describe nuclear
magnetic resonance measurements in pure and Y-doped single crystals. The
non-Kramers nature of the ground state doublet leads to a unique form of the
hyperfine coupling that exclusively probes the transverse field susceptibility.
Our results show that this quantity diverges at the critical field, in contrast
to the mean-field prediction. Furthermore, we find evidence for quantum
critical fluctuations present near Tm-rich regions in Y-doped crystals at
levels beyond which long-range order is suppressed, suggesting the presence of
quantum Griffiths phases.
