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The ground states of the frustrated pyrochlore oxideTb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ have been studied by inelastic neutron scatteringexperiments. Three single-crystal samples are investigated; one shows no phasetransition ($x=-0.007<x_{\text{c}}\sim -0.0025$), being a putative quantumspin-liquid (QSL), and the other two ($x=0.000, 0.003$) show electricquadrupole ordering (QO) below $T_{\text{c}} \sim 0.5$ K. The QSL sample showscontinuum excitation spectra with an energy scale 0.1 meV as well asenergy-resolution limited (nominally) elastic scattering. As $x$ is increased,pseudospin wave of the QO state emerges from this continuum excitation, whichagrees with that of powder samples and consequently verifies good $x$ controlfor the present single crystal samples.

Continuum excitation and pseudospin wave in quantum spin-liquid and quadrupole ordered states of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$