Vibronic collapse of ordered quadrupolar ice in the pyrochore magnet Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$

While the spin liquid state in the frustrated pyrochlore Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ has been studied both experimentally and theoretically for more than two decades, no definite description of this unconventional state has been achieved. Using synchrotron based THz spectroscopy in combination with quantum numerical simulations, we highlight a significant link between two previously unrelated features: the existence of a quadrupolar order following an ice rule and the presence of strong magneto-elastic coupling in the form of hybridized Tb$^{3+}$ crystal-field and phonon modes. The magnitude of this so-called vibronic process, which involves quadrupolar degrees of freedom, is significantly dependent on small off-stoichiometry $x$ and favors all-in all-out like correlations between quadrupoles. This mechanism competes with the long range ordered quadrupolar ice, and for slightly different stoichiometry, is able to destabilize it.