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The hidden ordered state of the frustrated pyrochlore oxideTb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ is possibly one of the two electric multipolar,or quadrupolar, states of the effective pseudospin-1/2 Hamiltonian derived fromcrystal-field ground state doublets of non-Kramers Tb$^{3+}$ ions. Theselong-range orders are antiparallel or parallel alignments of transversepseudospin components representing electric quadrupole moments, which cannot beobserved as magnetic Bragg reflections by neutron scattering. Howeverpseudospin waves of these states are composite waves of the magnetic-dipole andelectric-quadrupole moments, and can be partly observed by inelastic magneticneutron scattering. We calculate these spin-quadrupole waves using linearspin-wave theory and discuss previously observed low-energy magnetic excitationspectra of a polycrystalline sample with $x = 0.005$ ($T_{\text{c}} = 0.5 $ K).

Composite spin and quadrupole wave in the ordered phase of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$