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Spin correlations of the frustrated pyrochlore oxideTb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ have been investigated by using inelastic neutronscattering on single crystalline samples ($x=-0.007, 0.000,$ and $0.003$),which have the putative quantum-spin-liquid (QSL) or electric-quadrupolarground states. Spin correlations, which are notably observed in nominallyelastic scattering, show short-ranged correlations around $L$ points [$q =(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$], tiny antiferromagnetic Braggscattering at $L$ and $\Gamma$ points, and pinch-point type structures around$\Gamma$ points. The short-ranged spin correlations were analyzed using arandom phase approximation (RPA) assuming the paramagnetic state and two-spininteractions among Ising spins. These analyses have shown that the RPAscattering intensity well reproduces the experimental data using temperatureand $x$ dependent coupling constants of up to 10-th neighbor site pairs. Thissuggests that no symmetry breaking occurs in the QSL sample, and that a quantumtreatment beyond the semi-classical RPA approach is required. Implications ofthe experimental data and the RPA analyses are discussed.

Spin correlations of quantum-spin-liquid and quadrupole-ordered states of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$