Quadrupole Order in the Frustrated Pyrochlore Tb[2+x]Ti[2−x]O[7+y]

A hidden order that emerges in the frustrated pyrochlore Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ with $T_{\text{c}}=0.53$ K is studied using specific heat, magnetization, and neutron scattering experiments on a high-quality single crystal. Semi-quantitative analyses based on a pseudospin-1/2 Hamiltonian for ionic non-Kramers magnetic doublets demonstrate that it is an ordered state of electric quadrupole moments. The elusive spin liquid state of the nominal Tb$_2$Ti$_2$O$_7$ is most likely a U(1) quantum spin-liquid state.